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The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper provides an explicit construction of \(81\) symplectic resolutions of a \(4\)-dimensional quotient singularity obtained by an action of a certain symplectic group \(G\) of order \(32\) and also a construction of a new Kummer-type symplectic \(4\)-fold. For a complex vector space \(V\) with a symplectic form \(\omega\), denote by \(\mathrm{Sp}(V)\) its group of symplectomorphisms. An element \(A\in\mathrm{Sp}(V)\) is called a symplectic reflection if its fixed point set is of codimension \(2\). The paper studies the case when \(V=\mathbb{C}^4\), \(\omega=dx_1\wedge dx_3+dx_2\wedge dx_4\) in canonical coordinates, and the group \(G\) is generated by \(5\) symplectic reflections given by explicit \(4\times 4\) matrices (see Section~2.C of the paper). This group \(G\) has order \(32\) and is conjugate in \(\mathrm{Sp}(V)\) to the group generated by Dirac gamma matrices. So, the quotient singularity in question is \(Y=V/G\). A resolution \(\varphi: X\to Y\) is called symplectic, if \(X\) admits a symplectic form. The fact that \(Y\) has symplectic resolutions and that there are exactly \(81\) of them was proven earlier by \textit{G. Bellamy} and \textit{T. Schedler} [Math. Z. 273, No. 3--4, 753--769 (2013; Zbl 1271.16029)]. The new contribution of the article under consideration is an explicit construction of these resolutions. The first main result of the paper reads as follows. Let \(\mathbb{T}\) be a \(5\)-dimensional algebraic torus with coordinates \(t_i\), \(i=0,\dots,4\) associated to five classes of symplectic reflections generating \(G\). Let \(\mathcal{R}\) be a \(\mathbb{C}\)-subalgebra generated in \(\mathbb{C}[V]\otimes\mathbb{C}[\mathbb{T}]\) by \(t_{i}^{-2}\) for \(i=0,\dots,4\) and \(\phi_{ij}t_it_j\) for \(0\leq i<j\leq 4\), where \(\phi_{ij}\) are certain eigenfunctions of the action of the abelianization \(\mathrm{Ab}(G)\) on \(\mathbb{C}[V]^{[G,G]}\). Note that \(\mathrm{Ab}(G)\) can be identified with the class group \(\mathrm{Cl}(Y)\) and \(\mathbb{C}[V]^{[G,G]}\) with the Cox (or total coordinate) ring of \(Y\). Then there exist \(81\) GIT quotients of \(\mathrm{Spec}\mathcal{R}\) with respect to the action of an algebraic torus associated to a certain grading of \(\mathcal{R}\), and these quotients yield all symplectic resolutions of \(Y\). There is one distinguished resolution, and all other are obtained from it by flops. An interesting feature of the approach undertaken by the authors is the extensive use of the Cox ring. For the exact relation between the Cox ring of resolution of \(Y\) and the algebra \(\mathcal{R}\) the reader should consult the paper itself.
As an application of their technique, the authors describe an action of the group \(G\) on an abelian \(4\)-fold such that the resulting quotient admits a symplectic resolution and thus is a Kummer-type symplectic \(4\)-fold. Their second main result states that if \(\mathbb{E}\) is an elliptic curve with complex multiplication by \(\sqrt{-1}\), then there exists an embedding of the group \(G\) to the group \(\mathrm{Aut}(\mathbb{E}^4)\) such that the quotient \(\mathbb{E}^4/G\) has a resolution which is a Kummer symplectic \(4\)-fold \(X\) with \(b_2(X)=23\) and \(b_4(X)=276\).
The article is generally well written and self-contained. Some places are, however, computationally involved and the authors rely there on computer algebra systems as Macaulay2 and Singular. quotient singularity; symplectic resolution; Cox ring; symplectic manifold Donten-Bury, M.; Wiśniewski, J. A.: On 81 symplectic resolutions of a 4-dimensional quotient by a group of order 32. (2014) Global theory and resolution of singularities (algebro-geometric aspects), Minimal model program (Mori theory, extremal rays), Group actions on varieties or schemes (quotients), Geometric invariant theory, Divisors, linear systems, invertible sheaves, Hyper-Kähler and quaternionic Kähler geometry, ``special'' geometry On 81 symplectic resolutions of a 4-dimensional quotient by a group of order 32 | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be a finite subgroup of \(\operatorname{SL}(2,\Bbbk)\) and let \(R = \Bbbk[x,y]^G\) be the coordinate ring of the corresponding Kleinian singularity. In [Duke Math. J. 92, No. 3, 605--635 (1998; Zbl 0974.16007)], \textit{W. Crawley-Boevey} and \textit{M. P. Holland} defined deformations \(\mathcal{O}^\lambda\) of \(R\) parametrised by weights \(\lambda\). In this paper, we determine the singularity categories \(\mathcal{D}_{\operatorname{sg}}(\mathcal{O}^\lambda)\) of these deformations, and show that they correspond to subgraphs of the Dynkin graph associated to \(R\). This generalises known results on the structure of \(\mathcal{D}_{\operatorname{sg}}(R)\). We also provide a generalisation of the intersection theory appearing in the geometric McKay correspondence to a noncommutative setting. singularity categories; Kleinian singularities; preprojective algebras Singularities of surfaces or higher-dimensional varieties, Representations of quivers and partially ordered sets, Cohen-Macaulay modules in associative algebras, Derived categories, triangulated categories Singularity categories of deformations of Kleinian singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a nonsingular surface over complex numbers and \(S_n\) be the symmetric group on \(n\) letters. Denote the Hilbert scheme of \(n\) points on \(X\) by \(X^{[n]}\). By a result of Haiman \(X^{[n]}\) can be identified with the fine moduli space of \(S_n\)-clusters in \(X^n\). If we denote by \(\mathcal Z \subset X^{[n]}\times X^n\) the universal family of \(S_n\)-clusters and \(X^{[n]}\overset{q}{\leftarrow} \mathcal Z \overset{p}\rightarrow X^n\) be the projections then, the derived McKay correspondence of \textit{T. Bridgeland} et al. [J. Am. Math. Soc. 14, No. 3, 535--554 (2001; Zbl 0966.14028)] in this set up gives an equivalence of derived categories \(\Phi: D(X^{[n]})\overset{\sim}{\rightarrow} D_{S_n}(X^n)\) of (\(S_n\)-equivariant) coherent sheaves, where \(\Phi=Rp_*\circ q^*\).
Scala showed that for any vector bundle \(F\) on \(X\), the image of the tautological bundle \(F^{[n]}\) on \(X^{[n]}\) under \(\Phi\) is given by an explicit complex \(\mathsf C^\bullet_F\) of (\(S_n\)-equivariant) coherent sheaves concentrated in nonnegative degrees.
The paper under review studies the derived McKay correspondence above in the reverse order by means of \(\Psi=q_*^{S_n}\circ Lp^*\) (which is not the inverse of \(\Phi\)). The main result of the paper is that if one replaces \(\Phi\) by \(\Psi^{-1}\) the images of \(F^{[n]}\) and \(\bigwedge^k L^{[n]}\) where \(L\) is a line bundle are (explicitly given) sheaves (instead of complexes of sheaves).
This enables the author to prove new formulas and also give simpler proofs for existing formulas for homological invariants of tautological bundles and their wedge powers. derived McKay correspondence; Hilbert scheme; tautological bundle McKay correspondence, Parametrization (Chow and Hilbert schemes) Remarks on the derived McKay correspondence for Hilbert schemes of points and tautological bundles | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be a connected, simply-connected, simple algebraic group and let \(C\) be a smooth projective irreducible algebraic curve, the base field being the complex numbers. Let \({\mathfrak g}=\) Lie \(G\) and let \(R\) be the ring of regular functions on \(C - p\), where \(p\) is a fixed point of \(C\). The present article is devoted to an exposition of the relationship between
(a) the space of vacua of integrable highest weight representations of the affine Kac-Moody algebra \(\widehat {\mathfrak g}\) (or ``conformal blocks'' in the terminology of conformal field theory) and
(b) the space of regular sections of a line bundle on the moduli space \(\mathcal M\) of semistable principal \(G\)-bundles on \(C\) (or ``generalized theta functions'').
Let \(V\) be a finite dimensional representation of \(G\). The main goal of this survey is to present the following result, appeared in a paper by \textit{S. Kumar, M. S. Narashiman} and \textit{A. Ramanathan} [Math. Ann. 300, No. 1, 41-75 (1994; Zbl 0803.14012)]: For any \(d \geq 0\), the space \(H^{0}({\mathcal M}, \Theta(V)^{d})\) of regular sections of the so-called \(\Theta\)-bundle \(\Theta(V)\) and the space of \(g\otimes R\)-invariants in the full dual of certain irreducible highest weight module over \(\widehat g\) are isomorphic. The present exposition includes some preparatory material and some improvements in the proofs, compared with the paper cited above. The main result was also obtained by different methods by \textit{G. Faltings} [J. Algebr. Geom. 3, No. 2, 347-374 (1994; Zbl 0809.14009)] and, for \(G = SL(N)\), by \textit{A. Beauville} and \textit{Y. Lazlo} [Commun. Math. Phys. 164, No. 2, 385-419 (1994; Zbl 0815.14015)]. infinite Grassmannians; conformal blocks; Kac-Moody algebras; generalized theta functions; Verlinde formula; theta-bundle S. Kumar, ''An introduction to ind -varieties,'' Appendix B of ''Infinite Grassmannians and moduli spaces of \(G\)-bundles'' in Vector Bundles On Curves --.New Directions (Cetraro, Italy, 1995) , Lecture Notes in Math. 1649 , Springer, Berlin, 1997, 33--38. Theta functions and abelian varieties, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Algebraic moduli problems, moduli of vector bundles Infinite Grassmannians and moduli spaces of \(G\)-bundles | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper under review studies variations of moduli spaces of representations of preprojective \(K\)-algebras by applying tilting theory, to deal with minimal resolutions of Kleinian singularities.
To a finite subgroup \(G\subset SL(2,K)\) a McKay quiver \(Q\) is associated, and a \(K\)-algebra \(\Lambda\) associated to \(Q\), called preprojective algebra. Different resolutions of quotient singularities \(\mathbb{A}^{n}/G\) are encoded by different moduli spaces \(\mathcal{M} _{\theta,d}({\Lambda})\) of \(\theta\)-semistable \(\Lambda\)-modules of dimension vector \(d\), in the sense of \textit{A. D. King} [Q. J. Math., Oxf. II. Ser. 45, No. 180, 515--530 (1994; Zbl 0837.16005)]. Hence, by studying variations of moduli spaces for different stability parameters \(\theta\) we can study quotient singularities.
In section 2, tilting modules \(I_{w}\) (generalization in homological algebra of being torsion or torsion free for a module) are constructed over preprojective algebras. If we denote by \(\mathcal{S}_{\theta}(\Lambda)\subset Mod \Lambda\) the full subcategory of \(\theta\)-semistable \(\Lambda\)-modules, in section 3 relations between the moduli spaces are given, by showing that the functors \(Hom_{\Lambda}(I_{w},-)\) and \(-\otimes_{\Lambda}I_{w}\) induce equivalences between the categories \(\mathcal{S}_{\theta}(\Lambda)\) and \(\mathcal{S}_{w\theta}(\Lambda)\) (c.f. Theorem 3.13) which preserve \(S\)-equivalence classes, where \(w\) are elements of the Coxeter group. This induces a bijection between the sets of closed points in the moduli spaces, and in section 4 the equivalence can be extended to the respective derived categories to show that the bijection can be extended to an isomorphism of \(K\)-schemes (c.f. Theorem 4.20), by using the functors of points.
Section 5 is devoted to use the previous results in the framework of Kleinian singularities to generalize some results of \textit{W. Crawley-Boevey} (see, e.g., [Am. J. Math. 122, No. 5, 1027--1037 (2000; Zbl 1001.14001)]). In section 6 a full example is provided.
Note that the proofs work even in the case where \(Q\) is a non-Dynkin quiver with no loops. Combined with the homological nature of the proofs, this is why the authors expect to use the results in higher dimensions. moduli spaces; preprojective algebras; tilting theory; McKay correspondence; Kleinian singularities Sekiya, Y.; Yamaura, K., \textit{tilting theoretical approach to moduli spaces over preprojective algebras}, Algebr. Represent. Theory, 16, 1733-1786, (2013) Fine and coarse moduli spaces, Representations of quivers and partially ordered sets, McKay correspondence, Homological functors on modules (Tor, Ext, etc.) in associative algebras, Families, moduli, classification: algebraic theory Tilting theoretical approach to moduli spaces over preprojective 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 this important paper the authors apply methods of associative cluster tilting theory in order to investigate connections between Cohen-Macauley modules over 1-dimensional hypersurface singularities and the representation theory of associative algebras. One of the main result of the paper is Theorem 1.3. In this theorem the authors describe the number of rigid objects, basic cluster tilting objects, basic maximal rigid objects and indecomposable summands of basic maximal rigid objects in the stable category \(\underline{\mathrm{CM}}(R)\) of maximal Cohen-Macauley module over a simple one-dimensional singularity of dimension \(\geq 1\) over an algebraic closed field of characteristic 0.
Two proofs for this theorem are presented in Section 2 (using additive functions on the AR quiver) and in Section 3 (using the algorithm \texttt{Singular}). In the next section a large class of 1-dimensional hypersurface singularities having a cluster tilting object is constructed. In Section 5 and Section 6 connections between cluster tilting theory and birational geometry are exhibited. The main result concerning this subject are presented in Theorem 1.5 and Theorem 1.6. In the end of the paper the authors present an application to finite-dimensional algebras (Theorem 1.7). cluster tilting; 2-Calabi-Yau categories Ng, P.: A characterization of torsion theories in the cluster category of dynkin type \(A\)\_{}\{\(\infty\)\}. arXiv:1005.4364 Cohen-Macaulay modules, Representations of associative Artinian rings, Derived categories, triangulated categories, Rational and birational maps Cluster tilting for one-dimensional hypersurface singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0577.00010.]
In this article, the author gives a survey on the subject stated in the title developed by the author [Habilitationsschrift, Universität Bonn, 1984]. The author gives a definition (due to E. Looijenga) of an adjoint quotient for an arbitrary Kac-Moody Lie group G and analyses the structure of fibers. Due to Brieskorn, there is a relationship between simple singularities and simple algebraic groups. The author shows that at least to some extent there is a similar relationship between the deformation theory of simple elliptic and cusp singularities due to Looijenga and associated Kac-Moody Lie groups.
In the finite dimensional case, let G be a simply connected semi-simple algebraic group over \({\mathbb{C}}\), B be a Borel subgroup and \(T\subset B\) a maximal torus of G, N be the normalizer of T in G. Then, \(N/T=W\) is the finite Weyl group. The adjoint quotient of G is the quotient of G by its adjoint actions which is canonically isomorphic to T/W.
In the infinite-dimensional case, to obtain a reasonable adjoint quotient of G, the author uses the Tits cone attached to the root basis and its Weyl group and Looijenga's partial compactification of T/W. Its stratification into boundary components induces a partition of G which can be described in terms of the building associated with G.
Some open problems on a representation theoretic interpretation of this partition are mentioned at the end. Detailed proofs and relations to singularities may be found in the authors notes indicated above. Kac-Moody algebras; Kac-Moody group; deformation theory of singularities; adjoint quotient; Kac-Moody Lie group; semi-simple algebraic group; Borel subgroup; Weyl group; Tits cone Slodowy, P.: An adjoint quotient for certain groups attached to Kac-Moody algebras. Math. sci. Res. inst. Publ. 4, 307-333 (1985) Infinite-dimensional Lie groups and their Lie algebras: general properties, Singularities of curves, local rings, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras An adjoint quotient for certain groups attached to Kac-Moody algebras | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be a finite abelian group acting linearly on \(k^n\), where \(k\) is an algebraically closed field, and the order of \(G\) is invertible in \(k\) (so the action can be diagonalised). The present paper is a sequel to [Proc. Lond. Math. Soc. (3) 95, No. 1, 179--198 (2007; Zbl 1140.14046)], where the authors constructed expilitly an irreducible component \(Y_{\theta}\) (called the coherent component) of the moduli space of \(\theta\)-stable representations of the McKay quiver of \(G\).
Passing from a \(\theta\)-stable quiver representation to a \(G\)-constellation (i.e. a \(G\)-equivariant \(S\)-module isomorphic to the regular \(G\)-module \(kG\), where \(S\) is the coordinate ring of the \(G\)-module \(k^n\)), the authors can make use of Gröbner basis theory. They determine whether a given \(\theta\)-stable \(G\)-constellation corresponds to a point on the coherent component \(Y_{\theta}\). In the case when \(Y_{\theta}\) equals Nakamura's \(G\)-Hilbert scheme, they present explicit equations for a cover by local coordinate charts. The computational techniques introduced here are applied to construct a subgroup of \(GL(6,k)\) for which the \(G\)-Hilbert scheme is not normal. McKay quiver; Gröbner bases; \(G\)-Hilbert scheme Craw, A.; Maclagan, D.; Thomas, R.R., Moduli of mckay quiver representations II: Gröbner basis techniques, J. algebra, 316, 2, 514-535, (2007) Toric varieties, Newton polyhedra, Okounkov bodies, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Representations of quivers and partially ordered sets Moduli of McKay quiver representations. II: Gröbner basis techniques | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The goal of this paper is to illustrate on the simplest non-trivial example the use of representation theory to compute automorphic cohomology of arithmetic quotients \(\Gamma\backslash G/H\) of generalized period domains arising from Hodge theory, i.e. quotients of a homogeneous space \(G/H\) of a connected real Lie group \(G\) by an arithmetic subgroup \(\Gamma\subset G\). In the case under consideration, \(G=\mathrm{SL}(2, {\mathbb{R}})\), the space \({\mathcal H}=G/H\) is the upper half plane and \(Y_\Gamma=\Gamma\backslash{\mathcal H}\) is its quotient by a subgroup \(\Gamma\) of \(\mathrm{SL}(2,{\mathbb{Z}})\). Most of the time, for simplicity of exposition, \(\Gamma=\mathrm{SL}(2,{\mathbb{Z}})\).
To be able to do this, the author recalls the main facts of the representation theory of the group \(G=\mathrm{SL}(2,{\mathbb{R}})\) (finite dimensional representations, parabolic induction and principal series representations), introduces automorphic forms \({\mathcal A}(\Gamma,G)\) and Eisenstein series, modular forms and cuspidal automorphic forms \({}^0{\mathcal A}(\Gamma,G)\). The latter form a unitary \(G\)-submodule of \({\mathcal A}(\Gamma,G)\). Its decomposition as the (countable) algebraic sum of discrete series representations and unitary principal series representations is analyzed. It is later used to compute the Lie algebra cohomology \(H^*({\mathbf n}, {}^0{\mathcal A}(\Gamma,G))\) (here \({\mathbf n}={\mathbb{C}}{0\,0\choose 1\,0}\)), which turns out to be equal to the cuspidal automorphic cohomology of \(Y_\Gamma\).
The paper is an expanded version of lectures given by the author at a NSF/CBMS workshop ``Hodge theory, Complex Geometry, and Representation Theory'' (see [\textit{M. Green} et al., Mem. Am. Math. Soc. 1088, iii--v, 145 p. (2014; Zbl 1322.32017)]). Homogeneous spaces and generalizations, Representations of Lie and linear algebraic groups over real fields: analytic methods, Semisimple Lie groups and their representations, Homogeneous complex manifolds, Period matrices, variation of Hodge structure; degenerations Notes on the representation theory of \(\mathrm{SL}_2 (\mathbb R)\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Reid's recipe [\textit{M. Reid}, McKay correspondence, in: Proceedings of algebraic geom. symposium, Kinosaki 1997. 14--41 (1997), \url{arXiv:alg-geom/9702016}; \textit{A. Craw}, J. Algebra 285, No. 2, 682--705 (2005; Zbl 1073.14008)] for a finite abelian subgroup \(G\subset\text{ SL}(3;\mathbb{C})\) is a combinatorial procedure that marks the toric fan of the \(G\)-Hilbert scheme with irreducible representations of \(G\). The geometric McKay correspondence conjecture by \textit{S. Cautis} and \textit{T. Logvinenko} [J. Reine Angew. Math. 636, 193--236 (2009; Zbl 1245.14016)] that describes certain objects in the derived category of \(G\)-Hilbert in terms of Reid's recipe was later proved by \textit{T. Logvinenko} [J. Algebra 324, No. 8, 2064--2087 (2010; Zbl 1223.14018)] and \textit{S. Cautis} et al. [J. Reine Angew. Math. 727, 1--48 (2017; Zbl 1425.14013)]. We generalise Reid's recipe to any consistent dimer model by marking the toric fan of a crepant resolution of the vaccuum moduli space in a manner that is compatible with the geometric correspondence by \textit{R. Bocklandt} et al. [Math. Ann. 361, No. 3--4, 689--723 (2015; Zbl 1331.14022)]. Our main tool generalises the jigsaw transformations by \textit{I. Nakamura} [J. Algebr. Geom. 10, No. 4, 757--779 (2001; Zbl 1104.14003)] to consistent dimer models. Reid's recipe; dimer model; quiver moduli space; jigsaw transformations; tilting bundle McKay correspondence, Toric varieties, Newton polyhedra, Okounkov bodies, Derived categories and associative algebras, Representations of quivers and partially ordered sets Combinatorial Reid's recipe for consistent dimer models | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We propose a three dimensional generalization of the geometric McKay correspondence described by Gonzales-Sprinberg and Verdier in dimension two. We work it out in detail in abelian case. More precisely, we show that the Bridgeland-King-Reid derived category equivalence induces a natural geometric correspondence between irreducible representations of \(G\) and subschemes of the exceptional set of \(G\)-Hilb\((\mathbb C^3)\). This correspondence appears to be related to Reid's recipe. Cautis, S., Logvinenko, T.: A derived approach to geometric McKay correspondence in dimension three. J. Reine Angew. Math., 636, 193--236 (2009) McKay correspondence A derived approach to geometric McKay correspondence in dimension three | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this article, it is important that \(k\) is a fixed algebraically closed field of characteristic \(0\). The authors prove that the center of a homologically homogeneous, finitely generated \(k\)-algebra has rational singularities. Assume \(X=\text{Spec} R\) for an affine, Gorenstein \(k\)-algebra \(R\). In this article, a commutative resolution of singularities is a crepant homomorphism \(f:Y\rightarrow X,\) i.e. \(f^\ast\omega_Y=\omega_X.\) Bondal and Orlov conjectured that two such resolutions are derived equivalent, and this was later proved by Bridgeland. The authors generalize this to a \textit{third} noncommutative crepant resolution explaining Bridgeland's proof. This observation leads to different approaches to the Bondal-Orlov conjecture and related topics. The question now is how the existence of a noncommutative crepant resolution affects the original commutative singularity. It is known that if a Gorenstein singularity has a crepant resolution then it has rational singularities. The authors asks wether this is true for a noncommutative crepant resolution. The article answers this affirmatively.
Let \(\Delta\) be a prime affine \(k\)-algebra that is finitely generated as a module over its center \(Z(\Delta).\) \(\Delta\) is called homologically homogeneous of dimension \(d\) if all simple \(\Delta\)-modules have the same projective dimension \(d.\) The properties of homologically homogeneous rings are close to commutative regular rings, and the idea is to use such a ring \(\Delta\) as a noncommutative analogue of a crepant resolution. Formally, a noncommutative crepant resolution of \(R\) is any homologically homogeneous ring of the form \(\Delta=\text{End}_R(M)\) where \(M\) is a reflexive and finitely generated \(R\)-module. The main result of the article is the following:
Theorem. Let \(\Delta\) be a homologically homogeneous \(k\)-algebra. Then the center \(Z(\Delta)\) has rational singularities. In particular, if a normal affine \(k\)-domain \(R\) has a noncommutative crepant resolution, then it has rational singularities.
Also, examples are given proving that this theorem may fail in positive characteristic.
The article starts with the properties of homologically homogeneous rings, based on tame orders: If \(\Delta\) is a prime ring with simple Artinian ring of fractions \(A\) (i.e. \(\Delta\) is a prime order in \(A\)), \(\Delta\) is called a \textit{tame \(R\)-order} if it is a finitely generated and reflexive \(R\)-module such that \(\Delta_{\mathfrak p}\) is hereditary for all prime ideals \(\mathfrak p\) in \(R\) of height \(1\).
A homologically homogeneous ring \(\Delta\) of dimension \(d\) is Cohen Macaulay (CM) over its center \(Z(\Delta)\), both GK\(\dim\Delta\) and the global homological dimension gl\(\dim\Delta\) of \(\Delta\) equal \(d\), the center \(Z=Z(\Delta)\) is an affine CM normal domain, and finally, \(\Delta\) is a tame \(Z\)-order.
The rest of the article is then used to prove the main theorem. This involves reduction to the Calabi-Yau case for proving that \(Z\) has rational singularities by a generalization of the commutative method where one constructs a Gorenstein cover of a \(\mathbb Q\)-Gorenstein singularity.
The article ends with examples proving, among other things, that the main theorem may fail in the case where \(k\) has positive characteristic.
The article is precise, and illustrates noncommutative algebraic geometry in a concrete way. It also give useful criterions and ideas to be followed in other settings in noncommutative geometry. noncommutative crepant resolution; Rees ring; homologically homogeneous rings; tame orders J. T. Stafford and M. Van den Bergh, Noncommutative resolutions and rational singularities, Michigan Math. J. 57 (2008), 659-674. Special volume in honor of Melvin Hochster. Noncommutative algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Rings arising from noncommutative algebraic geometry, Homological dimension (category-theoretic aspects) Noncommutative resolutions and 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 Since Mumford's Geometric Invariant Theory quotients of actions of reductive groups and moduli spaces of vector bundles have become a major tool in classification theory. On the other hand representation theory of finite dimensional algebras and quivers provides a systematic study of classification problems which appear in many fields of mathematics. The basic work of \textit{A. D. King} [Q. J. Math., Oxf. II. Ser. 45, No. 180, 515-530 (1994; Zbl 0837.16005)] connects both fields: he shows the existence of the moduli space of representations over a finite dimensional algebra.
The author studies a class of moduli spaces which are closely related to preprojective algebras. After reviewing King's results he introduces framed moduli spaces and studies their properties, in particular their Grothendieck groups and their Chow rings. This is done by using a method of \textit{G. Ellingsrud} and \textit{S. A. Strømme} [J. Reine Angew. Math. 441, 33-44 (1993; Zbl 0814.14003)], and leads to an explicit description of generators for both groups as the classes of tautological bundles. Moreover these moduli spaces have the remarkable property, that the Chow ring and the singular cohomology ring coincide.
Further, the author describes the cotangent bundle as an open subvariety of the moduli space of representations of the corresponding preprojective algebra. Then he applies his results to simple singularities. He recalls McKay's construction and relates the moduli space of the preprojective algebra of a tame quiver to a quotient singularity of the form \({\mathcal C}/\Gamma\), where \(\Gamma\) is a finite subgroup of \(\text{SU}(2)\). Moreover he obtains a description of the exceptional set of the minimal resolution in terms of the quiver.
Finally, framed moduli are related to Kac-Moody algebras. A description of those algebras is given in terms of convolution algebras [\textit{V. Ginzburg}, C. R. Acad. Sci., Paris, Sér. I 312, No. 12, 907-912 (1991; Zbl 0749.17009)]. These convolution algebras are obtained from Lagrangian subvarieties of the product of framed moduli spaces of a fixed affine quiver with fixed frame, but various dimension vectors. Moreover those moduli spaces are ALE spaces [\textit{P. Kronheimer, H. Nakajima}, Math. Ann. 288, No. 2, 263-307 (1990; Zbl 0694.53025)], which are of interest in symplectic geometry. moduli spaces of vector bundles; finite dimensional algebras; quivers; Grothendieck groups; Chow rings; framed moduli; Kac-Moody algebras; convolution algebras Nakajima, H; Bautista, R (ed.); Martínez-Villa, R (ed.); Pena, JA (ed.), Varieties associated with quivers, No. 19, 139-157, (1996), Providence Representations of quivers and partially ordered sets, Families, moduli, classification: algebraic theory, Universal enveloping (super)algebras Varieties associated with quivers | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(\mathfrak{g}\) be a finite-dimensional complex simple Lie algebra with simply laced Dynkin diagram. Let \(N(\mathfrak{g})\) be the corresponding nilpotent cone and \(S\) the intersection of the transversal slice \(S_x\) of a subregular nilpotent element \(x\) with \(N(\mathfrak{g})\); this is a singular non-compact surface of the same type as \(\mathfrak{g}\).
It was studied intensively by Grothendieck, Brieskorn, Slodowy and others. In [Int. Math. Res. Not. 2014, No. 15, 4049--4084 (2014; Zbl 1312.14088)], the authors constructed ADE-bundles over these singularities, using the exceptional locus in its minimal resolution and bundle extensions.
In the present paper, they describe natural holomorphic filtrations of these bundles and various related ones. The main tool is the cohomology of line bundles over flag varieties and their cotangent bundles. ADE-singularities Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry, Special surfaces, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Structure theory for Lie algebras and superalgebras \(ADE\) bundles over \(ADE\) singular surfaces and flag varieties of \(ADE\) 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 The concept of a noncommutative crepant resolution (NCCR) of a singularity was introduced by Van den Bergh. An algebra acts as a substitute of an ordinary crepant resolution. This generalization stems from the fact that, for 3-dimensional terminal Gorenstein singularities, the derived category of representations of the substitute algebra is equivalent to the derived category of coherent sheaves of a commutative crepant resolution. The substitute algebra has the form \(A=\text{End}_R(T)\) where \(T\) is a reflexive \(R\)-module, and can be seen as a path algebra of a quiver \(Q\) with relations. The vertices corresponds to the direct summands of \(T\), and the set of arrows is a basic set of maps between them. Defining the dimension vector \(\alpha\) which assigns to a vertex the rank of the corresponding summand of \(T\), the singularity \(\text{Spec}(R)\) can be recovered as the moduli space parameterizing \(\alpha\)-dimensional semisimple representations, and often a commutative resolution can be constructed by constructing the moduli space of stable \(\alpha\)-dimensional representations, for some stability condition.
The author goes through the basic theory of crepant resolutions in a very nice way, so that the generalization to the noncommutative situation follows easily: A not necessarily commutative algebra \(A\) is a noncommutative crepant resolution of the (commutative) singularity \(R\) if
(1) \(A\cong\text{End}(T)\), where \(T\) is a finitely generated reflexive \(R\)-module
(2) \(A\) is homologically homogeneous, i.e., alle simple \(A\)-modules have the same projective dimension.
\noindent Van den Berg proved that, with respect to derived categories, in dimension 3, these algebras behave like crepant resolutions.
Now, in the 3-dimensional Gorenstein case, the NCCR's can be constructed using the concept of a \textit{maximal modification algebra} (MMA): An algebra \(A\) is called a modification algebra if it is of the form \(\text{End}(T)\) with \(T\) a reflexive module, and \(A\) is Cohen-Macaulay as an \(R\)-module. An NCCR is always an MMA, and in dimension 3, when \(R\) is a Gorenstein singularity, all NCCR's are MMA's.
To introduce the toric aspect, the author gives a very neat introduction to toric algebraic geometry. Then the noncommutative crepant resolutions can be obliged to carry an additional toric structure: All summands of \(T\) is supposed to have rank \(1\) and are graded (by the toric variety \(M\) in question). Then \(\text{End}(T)\) is called a toric noncommutative crepant resolution.
The interesting problem is to classify all possible toric noncommutative crepant resolutions of a given singularity. The author gives an algorithm to construct all such for any toric 3-dimensional Gorenstein singularity. This is (basically) done by using the algorithm to construct quivers \(Q\) that are also dimers, and then letting the algebra be the corresponding Jacobi algebra, which is the quiver algebra of \(Q\) with some particular ``dimer relations''. The toric conditions are strongly related to the representations of the quiver.
A method given by Craw and Quintero-Velez is used to embed the quivers of the toric NCCR algebras inside a real \(3\)-torus, such that the relations of the quiver algebra are precisely the homotopy relations. When the singularity is Gorenstein, the quiver can be projected to a 2-torus to obtain a dimer model.
The algorithm has to do with embedding the quivers into a torus, working on the varieties, and taking the corresponding quivers back to algebras (NCCR).
The algorithm is proven to be effective, in particular on the example of singularities from reflexive polygons and abelian quotients of the conifold. Then all dimer models corresponding to such a singularity are connected by mutations. Of course, generalizations of the algorithm to non toric, not Gorenstein, or higher dimensions are considered.
The article is very pedagogical, stringent, and it is a really good introduction to NCCR. quiver with relation; maximal modification algebra; noncommutative crepant resolution; toric geometry; graded rank 1 Cohen-Macaulay modules; dimer models; dimers; mutations R. Bocklandt, \textit{Generating toric noncommutative crepant resolutions}, arXiv:1104.1597 [INSPIRE]. Noncommutative algebraic geometry, Toric varieties, Newton polyhedra, Okounkov bodies, Derived categories, triangulated categories, Global theory and resolution of singularities (algebro-geometric aspects) Generating toric noncommutative crepant resolutions | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Following the author's previous works [Mich. Math. J. 54, No. 3, 517--535 (2006; Zbl 1159.14026); ibid. 62, No. 2, 253--363 (2006; Zbl 1322.14075)], this article addresses the derived McKay conjecture, stating an affirmative answer for finite abelian actions \(G\subset\)GL\((n,\mathbb{C})\) on \(\mathbb{C}^n\). More precisely, if \(f:Y\to X=\mathbb{C}^n/G\) is a \(\mathbb{Q}\)-factorial terminal relative minimal model of \(X\), then there exist toric closed subvarieties \(Z_i\subset X\) with \(Z_i\neq X\) for \(1\leq i\leq m\) and \(m\geq0\), such that there is a semi-orthogonal decomposition:
\[
D^b(coh([\mathbb{C}^n\!/G])) \cong \mathrm{Span}{D^b(coh(\widetilde{Z}_1)),\ldots,D^b(coh(\widetilde{Z}_m)),D^b(coh(\widetilde{Y}))}
\]
where \(\widetilde{Y}\) and \(\widetilde{Z}_i\) are smooth Deligne-Mumford stacks associates to \(Y\) and \(Z_i\) respectively. Moreover, if \(G\subset\)SL\((n,\mathbb{C})\) then \(m=0\) thus recovering the ``Classical'' derived McKay correspondence conjecture in this case.
For \(n=2\) the author is able to remove the abelian condition, so in this case it is proved that for any \(G\subset\)GL\((2,\mathbb{C})\) there is a semi-orthogonal decomposition:
\[
D^b(coh([\mathbb{C}^2\!/G])) \cong \mathrm{Span}{D^b(coh(Z^\nu_1)),\ldots,D^b(coh(Z^\nu_m)),D^b(coh(Y))}
\]
where \(Z^\nu_i\) are normalizations of \(Z_i\). If in addition the group \(G\) is small, i.e.\ it has no quasireflections, then the semi-orthogonal complement of \(D^b(coh(Y)\) in \(D^b(coh([\mathbb{C}^2\!/G]))\) is generated by an exceptional collection.
The paper concludes with the proof of the following result: Any proper birational morphism \(f:(X,B)\dashrightarrow(Y,C)\) between toric pairs which is a \(K\)-equivalence (i.e.\ \(K_X+B=K_Y+C\)), is decomposed into a sequence of flops. If in addition the coefficients of \(B\) belong to the estandar set \(\{1-1/m:m\in\mathbb Z_{>0}\}\) then \(D^b(coh(\widetilde{X})) \cong D^b(coh(\widetilde{Y}))\), thus proving in this case the ``\(K\) implies \(D\) conjecture'' proposed in [the author, J. Differ. Geom. 61, No. 1, 147--171 (2002; Zbl 1056.14021)]. McKay correspondence; toric variety; derived category; relative exceptional object Kawamata, Y., Derived categories of toric varieties III, European J. Math., 2, 196-207, (2016) McKay correspondence, Minimal model program (Mori theory, extremal rays), Toric varieties, Newton polyhedra, Okounkov bodies Derived categories of toric varieties. III | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 announces a method for solving a variation of the crepant resolution conjecture called the quantum McKay correspondence in the context of the disc invariants of the effective outer legs in the toric Calabi-Yau threefolds. The authors then establish the conjecture for the toric orbifold \([\mathbb{C}^3/\mathbb{Z}_5(1, 1, 3)]\) and its toric crepant resolution.
Inspired from the phase change phenomena in string theory, the main idea is to realize the given Calabi-Yau 3-orbifold and its toric crepant resolution as symplectic reductions of the same system of charge vectors. In this way, one can put the toric Calabi-Yau 3-orbifold and its toric crepant resolution in the same family. One can then use the charge vectors to compute genus zero closed Gromov-Witten invariants as well as the disc invariants with respect to Aganagic-Vafa branes, which are the open Gromov-Witten invariants as proposed by Brini and Cavalieri.
The paper under review also gives an enumerative explanation to the integrality of open-closed mirror maps in terms of Ooguri-Vafa invariants. quantum McKay correspondence; disc invariants; open mirror symmetry Ke, H.-Z., Zhou, J.: Quantum McKay correspondence for disc invariants of toric Calabi-Yau 3-orbifolds. arXiv:1410.4376 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Quantum McKay correspondence for disc invariants of toric Calabi-Yau 3-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 We give an introduction to the McKay correspondence and its connection to quotients of \(\mathbb {C}^n\) by finite reflection groups. This yields a natural construction of noncommutative resolutions of the discriminants of these reflection groups. This paper is an extended version of E. F.'s talk with the same title delivered at the ICRA. reflection groups; hyperplane arrangements; maximal Cohen-Macaulay modules; matrix factorizations; noncommutative desingularization McKay correspondence, Cohen-Macaulay modules, Global theory and resolution of singularities (algebro-geometric aspects), Noncommutative algebraic geometry Noncommutative resolutions of discriminants | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 0517.00010.]
This work is a short version of a paper which will be published by the author and \textit{J. L. Verdier} in Ann. Sci. Éc. Norm. Supér., IV. Sér. 16, 409-449 (1983; Zbl 0538.14033). The results are given without proofs. Let G be a finite subgroup of SL(2,\({\mathbb{C}})\) and S the surface obtained as the a quotient \(S={\mathbb{C}}^ 2/G\), having the origin as double rational singularity. If \(q:\widetilde S\to S\) is a minimal resolution of the singularity, D the exceptional fiber of q and Irr(D) the union of the irreducible components of D, there are well-known results of M. Artin connecting the group G and the dual graph \(\Gamma\) of Irr(D). In this paper are considered the ring R(G) of the representations of G and the Grothendieck ring \(K(\widetilde S)\) and are announced the following results: (1) The rings R(G) and \(K({\mathbb{C}}^ 2,0)\) are isomorphic. - (2) There is a bijection \(\pi *R(G)\to K(\tilde S)\). - (3) If \(c\in R(G)\) is the canonical representation there is a ring homomorphism \(R(G)/(2-c)R(G)\to K(\tilde S)/K_ D(\tilde S)\). Grothendieck ring; representations of group of automorphisms; double rational singularity Singularities of surfaces or higher-dimensional varieties, Group actions on varieties or schemes (quotients), Representation theory for linear algebraic groups, Homogeneous spaces and generalizations, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) Representation of polyhedral groups and singularities of surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a smooth algebraic variety over a field \(k\) and \(G\) be a finite group acting on \(X\) (such that \(| G| \) does not divide the characteristic of \(k\)). Let \(Y\) be a resolution of the singular quotient variety \(X/G\). Establishing links between geometric objects on \(Y\) and \(G\)-equivariant objects of the same kind on \(X\) is usually referred to as McKay-correspondence.
In this context for \(X/G\) Gorenstein and \(\pi:Y\to X/G\) a crepant resolution of singularities, Ruan's cohomological conjecture states that \(H^*(Y,{\mathbb C})\) should be isomorphic to the orbifold cohomology of the couple \((X,G)\).
The author takes a categorical approach to this conjecture: the goal is to extract it from an equivalence \(D^b(Y)\to D^b_G(X)\) between the bounded derived categories of coherent sheaves on \(Y\) and the one of \(G\)-equivariant sheaves on \(X\). The article under review works with the corresponding categories of vector bundles Vect\((Y)\) and of \(G\)-equivariant vector bundles Vect\(_G(X)\). The former equivalence holds true in dimension \(\leq 3\) as shown by Bridgeland-King-Reid in dimension \(3\) using the resolution by the Hilbert scheme \(Y=G-\text{Hilb}(X)\) parametrising \(G\)-clusters in \(X\) [\textit{T. Bridgeland}, \textit{A. King} and \textit{M. Reid}, J. Am. Math. Soc. 14, No. 3, 535--554 (2001; Zbl 0966.14028)].
The main idea here is to use \textit{B. Keller}'s construction [J. Pure Appl. Algebra 136, No. 1, 1--56 (1999; Zbl 0923.19004)] of a mixed complex \(C({\mathcal A})\) leading to homology theories \(HC_{\bullet}({\mathcal A},W)\) associated to an exact category \({\mathcal A}\) and a graded \(k[u]\)-module \(W\) of finite projective dimension. For special choices of \(W\) one recovers Hochschild, cyclic, periodic cyclic and negative cyclic homology. Let \(G\) act on the disjoint sum of fixed point sets \(\coprod_{g\in G}X^g\) such that \(h\in G\) sends \(x\in X^g\) to \(h\cdot x\in X^{hgh^{-1}}\). This action is inherited by \(\bigoplus_{g\in G}HC_{\bullet}(\text{Vect}(X^g),W)\). The main theorem reads:
Let \(G\) be a finite group acting on a smooth quasiprojective variety \(X\) over a field \(k\) of characteristic not dividing \(| G| \). For any graded \(k[u]\)-module \(W\) there exists an isomorphism functorial with respect to pullbacks under \(G\)-equivariant maps:
\[
\psi_X:HC_{\bullet}(\text{Vect}_G(X),W)\simeq\biggl(\bigoplus_{g\in G}HC_{\bullet}(\text{Vect}(X^g),W)\biggr)_G
\]
where \((\ldots)_G\) denotes the \(G\)-coinvariants.
From the theorem it follows rather straightforwardly that an equivalence \(D^b(Y)\to D^b_G(X)\) implies Ruan's conjecture. The earliest version of this type of theorem seems to go back to \textit{B. L. Feigin} and \textit{B. L. Tsygan} [in: \(K\)-theory, arithmetic and geometry. Lect. Notes Math. 1289, 67--209 (1987; Zbl 0635.18008)].
The proof occupies \(12\) pages and is mainly a sequence of reduction steps: first of all, Keller's construction is taken relatively with respect to the subcategory of bounded acyclic complexes. These relative mixed complexes are denoted by \(C({\mathcal Ac}^b(Y),{\mathcal C}^b(Y))=:C(Y)\) resp. \(C({\mathcal A}c^b_G(X),{\mathcal C}_G^b(X))=:C_G(X)\), where \({\mathcal C}^b(Y)\) is the category of bounded complexes of vector bundles on \(Y\). Mixed complexes are regarded as \(\Lambda\)-modules for some DG \(k\)-algebra \(\Lambda\), and for the claim of the above theorem, it is enough to consider them as elements of the derived category of \(\Lambda\)-modules. The claim reduces then to a quasiisomorphism of the \(\Lambda\)-modules \(C_G(X)\) and \((\bigoplus_{g\in G}C(X^g))_G\).
Further, \(C_G(X)\) can be replaced by Keller's construction on the corresponding ``crossed product categories'': \({\mathcal C}^b(X)\rtimes G\) is the full subcategory of \({\mathcal C}^b_G(X)\) formed by objects of the type \(\widetilde{\mathcal F}=\bigoplus_{g\in G}g{\mathcal F}\) with the natural \(G\)-equivariant structure, for some complex of vector bundles \({\mathcal F}\). With this replacement, the map underlying the wanted quasiisomorphism is constructed in a natural way.
A Mayer-Vietoris argument reduces the claim that the so constructed map is a quasiisomorphism to an affine variety \(X=\text{Spec}(A)\). More precisely, let \(J_g\subset A\) denote the ideal corresponding to the fixed point set \(X^g\subset X\), then we are reduced to showing that
\[
\psi_A:C(A\rtimes G)\to\biggl(\bigoplus_{g\in G}C(A/J_g)\biggr)_G
\]
defined by
\[
\psi(a_0\cdot g_0,\ldots,a_n\cdot g_n)\,=\,\overline{(a_0,g_0(a_1),\ldots,(g_0\ldots g_{n-1})(a_n))}
\]
(where the bar denotes the class in the space of coinvariants) is a quasiisomorphism, where \(A\rtimes G\) is now the usual crossed product algebra.
\textit{E. Getzler} and \textit{J. D. S. Jones} [J. Reine Angew. Math. 445, 161--174 (1993; Zbl 0795.46052)] showed that \(C(A\rtimes G)\) is quasiisomorphic to some mixed complex \(C(\bigoplus_{g\in G}A^{\sharp}_g)_G\) constructed explicitely from cyclic simplicial data \((A^{\sharp}_g)_n=A^{\otimes(n+1)}\) \(\forall n\in{\mathbb N}\) with face, degeneracy maps and cyclic operators. Thus it remains to see that \(C(\bigoplus_{g\in G}A^{\sharp}_g)_G\to(\bigoplus_{g\in G}C(A/J_g))_G\) (induced by the obvious quotient map) is a quasiisomorphism.
Identifying \(\bigoplus_{g\in {\mathcal O}}A^{\sharp}_g\) and \(\bigoplus_{g\in {\mathcal O}}C(A/J_g)\) for a conjugacy class \({\mathcal O}\subset G\) as induced modules from the centralizer \(C_g\) of \(g\), Shapiro's lemma reduces the claim to a quasiisomorphism
\[
C(A^{\sharp}_g)_{C_g}\simeq C(A/J_g)_{C_g}.
\]
Luna's fundamental lemma allows to show the existence of an affine \(G\)-equivariant covering of \(X\) isolating the fixed points and flattening the situation up to \(G\)-equivariant étale morphism. Étale descent then reduces the situation to a polynomial ring \(A\) with an action of a cyclic group \(\langle g\rangle\). This case is finally resolved using Koszul resolutions.
The article closes with some explicit cases (such as \(G-\text{Hilb}(X)\) as a resolution of \(X/G\)) and conjectures about the identification of the product structure on cohomology. derived categories; crepant resolution; mixed complex V. Baranovsky, Orbifold cohomology as periodic cyclic homology. Internat. J. Math. 14 (2003), 791-812. Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Classical real and complex (co)homology in algebraic geometry, Generalizations (algebraic spaces, stacks), Derived categories, triangulated categories, \(K\)-theory and homology; cyclic homology and cohomology Orbifold cohomology as periodic cyclic homology. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 continues a line of research which, as the first approximation, can be drawn as follows. In [\textit{A. Bondal} and \textit{D. Orlov}, in: Proceedings of the international congress of mathematicians, ICM 2002, Beijing, China, August 20--28, 2002. Vol. II: Invited lectures. Beijing: Higher Education Press; Singapore: World Scientific/distributor. 47--56 (2002; Zbl 0996.18007)], the authors conjectured that an appropriate theory of non-commutative resolution could augment the usual commutative resolution. [\textit{M. van den Bergh}, in: The legacy of Niels Henrik Abel. Papers from the Abel bicentennial conference, University of Oslo, Oslo, Norway, June 3--8, 2002. Berlin: Springer. 749--770 (2004; Zbl 1082.14005)], proposed a definition of a non-commutative crepant resolution of Gorenstein singularities. [\textit{R. Bocklandt}, J. Algebra 364, 119--147 (2012; Zbl 1263.14006)], described how the dimer models and the corresponding quivers can be used to construct non-commutative crepant resolutions of \(3\)-dimensional Gorenstein toric singularities. Such a resolution is obtained as the endomorphism ring of a certain module which, in general, is not unique. The present paper investigates the relationship between the modules giving non-commutative crepant resolutions of the same singularity.
For the purposes of this review, the following definition should suffice. Let \(R\) be a \(d\)-dimensional Cohen-Macaulay local ring and \(M\) a reflexive module over \(R\). The \(R\)-algebra \(\Lambda= \mathrm{End}_R\,M\) is called a \textit{non-commutative crepant resolution (NCCR)} of \(R\) if the depth of \(\Lambda\) as an \(R\)-module is \(d\) and \(\mathrm{gl.dim}\,\Lambda_\mathfrak{p}= \dim R_\mathfrak{p}\) for all \(\mathfrak{p}\in\mathrm{Spec}\,R\). A \textit{dimer model} is a finite polygonal decomposition of the \(2\)-dimensional real torus. Note that the first homology group of the torus is \(\mathbb{Z}^2\). Let us view this \(\mathbb{Z}^2\) as the ``hight \(1\)'' plane \(z=1\) in the cocharacter lattice \(\mathbb{Z}^3\) of the \(3\)-dimensional algebraic torus \((\mathbb{C}^*)^3\). Under certain convenient conditions on the dimer model, it also produces a lattice polygon \(\Delta\) in \(\mathbb{Z}^2 \otimes\mathbb{R}\); considering the cone \(\sigma_\Delta\subset \mathbb{Z}^3 \otimes\mathbb{R}\) over this polygon, one gets a \(3\)-dimensional Gorenstein toric singularity. The paper under consideration focuses on the \textit{reflexive} polygons, i.e., lattice polygons \(\Delta\) such that the origin is the only integral point in the interior of \(\Delta\).
In a dual way, one can also associate a quiver \(Q(\Gamma)\) to a dimer model \(\Gamma\) and define a distinguished element \(W_Q\) called a \textit{superpotential} of the path algebra of \(Q(\Gamma)\). In turn, one defines the \textit{Jacobian} (or \textit{superpotential}) \textit{algebra} \(\mathcal{P}(Q,W_Q)\) as a certain quotient of the path algebra. It is quite not obvious, but can be proven, that \(\mathcal{P}(Q,W_Q)\) has the form \(\mathrm{End}_R\, M\) for some module \(M\) (which can also be read from the combinatorial data) and provides a NCCR of \(R\). Moreover, \(R\) is the center of \(\mathcal{P}(Q,W_Q)\). This is one of Bocklandt's results; he also descibed a combinatorial procedure called \textit{mutation} which transforms one quiver with potential to another one. Bocklandt also proved that if \(R\) is a \(3\)-dimensional Gorenstein toric singularity associated with a reflexive polygon, then any two consistent dimer models giving splitting NCCRs of \(R\) are connected by a sequence of mutations. It must also be added that it is known that each \(3\)-dimensional Gorestein toric singularity as well as its NCCR can be obtained from a dimer model and each such singularity has only finitely many NCCRs. These results should be compared with the common commutative small resolution of a \(3\)-dimensional Gorenstein toric singularity where each pair of resolutions is connected by a sequence of flops.
On the other hand, the modules \(M\) providing the NCCR (in the form \(\mathrm{End}_R\, M\); they are called \textit{splitting maximal modifying (MM) generators}) are also not unique. The procedure of mutation extends to these splitting MM generators. Now the main result of the paper states that, for a given reflexive polygon \(\Delta\) and the corresponding \(3\)-dimensional Gorenstein toric singularity \(R\), the set of splitting MM generators is connected via mutations. The proof is based on the known classification of reflexive polygons and goes by a direct calculation of splitting MM generators and their mutations in each case.
The paper starts with preliminary material; this comprises around \(1/4\) of its volume. The rest \(3/4\) consists of the case by case proof of the main result. The author definitely tried to make the paper self-contained. The sections on dimer models and quivers are more or less elementary, still the extracts from the general theory of non-commutative resolution are less accessible to a non-expert. Such a reader can be advised to consult (as the reviewer did) an excellent survey of \textit{G. J. Leuschke} [``Non-commutative crepant resolutions: scenes from categorical geometry'', Preprint, \url{arXiv:1103.5380}]. non-commutative crepant resolution; dimer model; quiver with potential; Gorenstein toric singularity; mutations of dimer models Noncommutative algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Toric varieties, Newton polyhedra, Okounkov bodies Mutations of splitting maximal modifying modules: the case of reflexive polygons | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author considers a finite cyclic group \(G\subset \text{GL}(2,\mathbb C) \) acting freely on \(\mathbb C^2\setminus \{ 0 \} \) and studies the singularity \(X= \mathbb C^2/ G\). The main theorems are:
Let \(I_{G}\) be an ideal of \({ \mathcal O }_{{ \mathbb C^2 }}\) defined by the free \(G\)-orbit. Then the Gröbner fan for the \(G\)-homogeneous ideal \(I_{G}\) determines a toric variety which is isomorphic to the minimal resolution of \(X\).
There is a bijection between irreducible special representations \(\rho_{k } \) of \(G\) and binomial generators in the initial ideal \(in_{w(I)}\).
This can be interpreted as a generalized MacKay correspondence. cyclic quotient singularities; Gröbner fan; MacKay correspondence Ito, Y.: Minimal resolution via Gröbner basis. Algebraic geometry in east Asia (Kyoto, 2001), 165-174 (2002) Global theory and resolution of singularities (algebro-geometric aspects), Parametrization (Chow and Hilbert schemes), Toric varieties, Newton polyhedra, Okounkov bodies Minimal resolution via Gröbner basis | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 somewhat build as an article. This means that the expert on the field could read the introduction to the book, then knowing what the author is doing. In the later chapters of the book all details are given thoroughly. The book explains the theory of Cayley smooth orders in central simple algebras over function fields of varieties. It describes the étale local structure of such orders as well as their central singularities and finite dimensional representations. This is motivated by two reasons: The first application is the construction of partial desingularizations of singularities from noncommutative algebras. The second motivation stems from noncommutative algebraic geometry. One attempts to study formally smooth algebras, or quasi-free algebras via their finite dimensional representations which, in turn, are controlled by associated Cayley-smooth algebras.
The first four chapters includes background information on a variety of topics including invariant theory, algebraic geometry, central simple algebras and the representation theory of quivers. Then chapters 5 and 6 contain the main results on Cayley-smooth orders. The author describes the étale local structure of a Cayley smooth order in a semisimple representation and classify the associated central singularity up to smooth equivalence. This is done by associating to a semisimple representation a combinatorial gadget called a \textit{marked quiver setting}, which encodes the tangent-space information to the noncommutative manifold in the cluster of points determined by the simple factors of the representation.
Let \(G\) be a finite group acting on \(\mathbb{C}^2\) free away from the origin. Resolutions \(Y\rightarrow\mathbb{C}^d/G\) have been constructed using the skew group algebra \(\mathbb{C}[x_1,\dots,x_d]\sharp G\) which is an order with center \(\mathbb{C}[\mathbb{C}^2/G]=\mathbb{C}[x_1,\dots,x_D]^G\) or deformations of it. In dimension \(d=2\) this gives minimal resolutions via the connection with the preprojective algebra, in dimension \(d=3\) the skew group algebra appears via the superpotential and commuting matrices setting. or via the McKay quiver. If \(G\) is abelian one obtain crepant resolutions, but for general \(G\) one obtains at best partial resolutions with conifold singularities remaining. In dimension \(d>3\) the situation is unclear. One goal of this text is to find a noncommutative explanation for the omnipresence of conifold singularities in partial resolutions of three-dimensional quotient singularities. The book concludes (already in the introduction) with a suggestion of a list of nice singularities in higher dimensions. This list comes from the hope that any quotient singularity \(X=\mathbb{C}^d/G\) has associated to it a nice order \(A\) with center \(R=\mathbb{C}[X]\) such that there is a stability structure \(\theta\) such that the scheme of all \(\theta\)-semistable representations of \(A\) is a smooth variety. If this is the case, the associated moduli space will be a partial resolution
\[
\text{moduli}^\theta_\alpha A\twoheadrightarrow X=\mathbb{C}^d/G
\]
and has a sheaf of Cayley-smooth orders \(\mathcal{A}\) over it, allowing us to control its singularities in a combinatorial way. If \(a\) is a Cayley-smooth order over \(R=\mathbb{C}[X]\) then its noncommutative variety \(\text{max} A\) of maximal twosided ideals is birational to \(X\) away from the ramification locus. If \(P\) is a point of the ramification locus \(\text{ram} A\) then there is a finite cluster of infinitesimally nearby noncommutative points lying over it. The local structure of the noncommutative variety \(\text{max} A\) near this cluster can be summarized by a marked quiver setting \((Q,\alpha),\) wich in turn allows to compute the étale local structure of \(A\) and \(R\) in \(P\). The central singularities appearing this way have been classified, giving the small lists of singularities.
The book includes noncommutative algebra, that is central simple algebras, \(R\)-orders, smoothness (Serre and Cayley), Grothendieck topology, Zariski twisted forms, Azumaya algebras, the Brauer group, Cayley-Hamilton algebras and the \(PGL_n\)-scheme \(\text{trep}_n\) classifying all trace preserving \(n\)-dimensional representations \(A\overset\phi\rightarrow M_n(\mathbb{C})\) of \(A\).
Noncommutative geometry in this book associates to \(A\in\text{alg}_n\) the noncommutative variety \(\text{max} A\) and argue that this gives a noncommutative manifold when \(A\) is a Cayley-smooth order. Notice the one new feature which noncommutative geometry has to offer compared to commutative geometry: Distinct points can lie infinitesimally close to each other. As desingularizations is the process of separating bad tangents, this fact is used in this noncommutative geometry. A central point in this theory is the marked quiver setting \((Q^{\bullet},\alpha,\beta)\) where \(\alpha,\beta\) are the Bergman-Small data.
Chapter 1 discusses the category \(\text{alg}_n\) of degree \(n\) Cayley-Hamilton algebras. This includes a thorough research on the classification of conjugacy classes of matrices in several settings. This includes trace and necklace relations. Chapter 2 associates to an affine \(\mathbb{C}\)-algebra \(A\) its affine scheme of \(n\)-dimensional representations \(\text{rep}_n A\) and study this scheme under the action of \(GL_n\). This chapter proves the Hilbert criterium and Artin's result, and also gives the necessary introduction to geometric invariant theory.
Chapter 3 called Étale Technology generalizes this technology to noncommutative geometry. Étale cohomology groups are used to classify central simple algebras over function fields of varieties. Orders in such central simple algebras are an important class of Cayley Hamilton algebras. This book introduce the class of \textit{Cayley-smooth orders}, which does allow an étale local description in arbitrary dimensions. This chapter investigates étale slices of representation varieties at semisimple representations. Notice that this includes ramification divisors, Knop-Luna slices and several spectral sequences.
Chapter 4 on quivers defines Cayley-smooth algebras \(A\in{\text{alg}}_n\) which are analogous to smooth commutative algebras. It is proved that the local structure of \(A\) in a point \(\xi\in\text{triss}_n A\) is determined by a marked quiver \((Q,\alpha).\) Chapter 5 about semisimple representations extends some results on quotient varieties of representations of quivers to the setting of marked quivers. A computational method is given to verify wether \(\xi\) belongs to the Cayley-smooth locus of \(A\). In low dimensions a complete classification of all marked quiver settings that can arise for a Cayley-smooth order is given.
Chapter 6 is the study of nilpotent representations. That is the study of the fibers of the quotient map
\[
\text{trep}_n A\overset\pi\twoheadrightarrow{\text{triss}}_n A.
\]
If \((Q^\bullet,\alpha)\) is the local marked quiver setting of a point \(\xi\in{\text{triss}}_n A\) then the \(GL_n\)-structure of the fiber \(\pi^{-1}(\xi)\) is isomorphic to the \(GL(\alpha)\)-structure of the nullcone \(\text{Null}_\alpha Q^\bullet\) consisting of all nilpotent \(\alpha\)-dimensional representations of \(Q^\bullet\). In GIT, nullcones are investigated by a refinement of the Hilbert criterion, the Hesselink stratification. The main aim of this chapter is to prove that the different strata in the Hesselink stratification of the nullcone of quiver-representations can be studied via moduli spaces of semistable quiver-representations.
Chapter 7 constructs noncommutative manifolds, that is families \((X_n)_n\) of commutative varieties that are locally controlled by Quillen-smooth algebras. Chapter 8 studies the application to families \((Y_n)_n\) where the role of Quillen-smooth algebras is replaced by Cayley-smooth algebras and where the sum-maps are replaced by gluing into a larger space. This final chapter gives the details of Ginzburg's coadjoint-orbit result for Calogero-Moser phase space which was the first instance of such a situation.
The final comment is then that this is a really entertaining book, covering most of the topics on noncommutative geometry. Also it is reasonable elementary, easy to read, easy to understand, and if the reader would like to go further into details, an extensive bibliography is given. However, we observe that there are few references to formal methods. Le Bruyn, L.: Noncommutative geometry and Cayley-smooth orders. In: Pure and Applied Mathematics (Boca Raton), 290. Boca Raton, FL: Chapman \& Hall/CRC, 2008 Noncommutative algebraic geometry, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Research exposition (monographs, survey articles) pertaining to associative rings and algebras Noncommutative geometry and Cayley-smooth orders | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We develop, for finite groups, a certain kind of resolutions of quotient spaces \(\mathbb{C}^n/G\) called pluri-toric resolutions. The pluri-toric resolutions extend the use of toric geometry, from the study of quotients by abelian finite groups to the study of quotients by arbitrary finite groups. This is done in such a way that the combinatorical and stratifying nature of toric resolutions is inherited by the pluri-toric resolutions.
We show that some of the properties of a pluri-toric resolution of \(\mathbb{C}^n/G\) can be deduced directly from the data of the group \(G\). One of these results states that a pluri-toric resolution, of a quotient by a finite subgroup of \(SL(n,\mathbb{C})\) where \(n\) is 2 or 3, is always crepant. The pluri-toric resolutions also give a generalised degree one McKay correspondence, i.e. a correspondence between certain conjugacy classes in \(G\) and the exceptional prime divisors in a pluri-toric resolution of \(\mathbb{C}^n/G\).
The pluri-toric resolutions are constructed in two steps. In the first step we use a toric resolution of the quotient space \(\mathbb{C}^n/A\), where \(A\) is a maximal abelian subgroup of \(G\), to construct a partial resolution, called a mono-toric partial resolution, of \(\mathbb{C}^n/G\). In the second step we take a mono-toric partial resolution for each maximal abelian subgroup of \(G\) and patch these together to a pluri-toric resolution of \(\mathbb{C}^n/G\). Even if the construction addresses the general case we mainly focus on the cases when \(\mathbb{C}^n/G\) is a surface or a threefold.
Our treatment leaves a number of open questions. resolutions of quotient spaces; pluri-toric resolutions; McKay correspondence Global theory and resolution of singularities (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies, Equisingularity (topological and analytic), Homogeneous spaces and generalizations Pluri-toric resolutions of quotient singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities For every bounded Hermitian symmetric domain \(\Omega\) and every near arithmetic subgroup \(\Gamma\) of \(\Aut \Omega\) the quotient \(X: = \Gamma \backslash \Omega\) is a quasiprojective manifold and admits a smooth (minimal) toroidal compactification \(\overline X\) [see \textit{A. Ash}, \textit{D. Mumford}, \textit{M. Rapoport} and \textit{Y. Tai}, `Smooth compactification of locally symmetric varieties' (1975; Zbl 0334.14007) and \textit{Y. Namikawa}, `Toroidal compactification of Siegel spaces' (1980; Zbl 0466.14011)].
The author studies the structure of the boundary divisor \(D: = \overline X \backslash X\) and the canonical bundle \(K_{\overline X}\) of \(\overline X\) for \(\Omega: = S_ n\), the Siegel upper half space of rang \(n\), and \(\Gamma: = \Gamma (k)\), \(k \geq 3\), a (neat) principal congruence subgroup of \(Sp (n,\mathbb{Z})\). He gets precise and detailed results in the case \(n=2\), in particular for the canonical bundle \(K_{\overline X}\) (Theorem 3.1) and for the description of the singular canonical volume form of \(\overline X\) along \(D\) (Theorem 4.1). Furthermore he shows that \(\Gamma (k) \backslash S_ 2\) is of general type for \(k \geq 4\) (Theorem 3.2). locally symmetric Hermitian space; toroidal compactification Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects), Compactification of analytic spaces, Toric varieties, Newton polyhedra, Okounkov bodies On the smooth compactification of Siegel 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 We study variations of tautological bundles on moduli spaces of representations of quivers with relations associated with dimer models under a change of stability parameters. We prove that if the tautological bundle induces a derived equivalence for some stability parameter, then the same holds for any generic stability parameter, and any projective crepant resolution can be obtained as the moduli space for some stability parameter. This result is used in [the authors, Geom. Topol. 19, No. 6, 3405--3466 (2015; Zbl 1338.14019)] to prove the abelian McKay correspondence without using the result of [\textit{T. Bridgeland} et al., J. Am. Math. Soc. 14, No. 3, 535--554 (2001; Zbl 0966.14028)]. dimer model; toric Calabi-Yau 3-fold; variation of GIT Ishii, A., Ueda, K.: Dimer models and crepant resolutions. To appear in Hokkaido Mathematical Journal (2013). arXiv:1303.4028 Algebraic moduli problems, moduli of vector bundles, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), McKay correspondence, Calabi-Yau manifolds (algebro-geometric aspects) Dimer models and crepant resolutions | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Triangulated categories of singularities may be seen as a categorical measure for the complexity of the singularities of a Noetherian scheme \(X\). For a commutative Noetherian ring \(R\), the category \(\mathcal D_{sg}(R)=\frac{\mathcal D^b(\mathrm{mod}-R)}{\mathrm{Perf}(R)}\), where \(\mathrm{Perf}(R)\) is the subcategory of perfect sheaves, i.e., complexes which are quasi-isomorphic to bounded complexes of finitely generated projective modules, is called the \textit{singularity category} of \(R\)
If \(X\) has isolated Gorenstein singularities \(x_1,\dots, x_n\), it is known that the triangulation of the singularity category is equivalent to the direct sum of the stable categories of maximal Cohen-Macaulay \(\widehat{\mathcal O}_{x_i}\)-modules.
\textit{M. Van den Bergh} [Duke Math. J. 122, No. 3, 423--455 (2004; Zbl 1074.14013)] defined noncommutative analogues of (crepant) resolutions (NC(C)R) of singularities which gives pure commutative results. Also, moduli spaces of quiver representations give useful techniques to obtain commutative resolutions from noncommutative ones.
Combining these concepts, the first author together with \textit{I. Burban} [Adv. Math. 231, No. 1, 414--435 (2012; Zbl 1249.14004)] developed the theory of \textit{relative singularity categories}. The idea is that these categories measure the difference between the derived category of a noncommutative resolution (NCR) and the smooth part \(K^b(\mathrm{proj}-R)\subseteq D^b(\mathsf{mod}-R)\) of the derived category of the singularity. This article is about the relation between relative and classical singularity categories.
Using the theories developed here, the authors obtain a purely commutative result. This is in cooperation with Iyama and Wemyss [\textit{M. Kalck} et al., Compos. Math. 151, No. 3, 502--534 (2015; Zbl 1327.14172)], and consists of decomposing \textit{O. Iyama} and \textit{M. Wemyss}' \textit{new triangulated category} [Ill. J. Math. 55, No. 1, 325--341 (2011; Zbl 1258.13015)] for complete rational surface singularities into blocks of singularities categories of ADE singularities. Moreover, \textit{L. Thanhoffer de Völcsey} and \textit{M. Van den Bergh} [``Explicit models for some stable categories of maximal Cohen-Macaulay modules'', Preprint, \url{arXiv:1006.2021}] have proved that the stable category of a complete Gorenstein quotient singularity of Krull dimension 3 is a generalized cluster category. In this article, the authors recover these results using different techniques.
Let \(k\) be an algebraically closed field, let \((R,\mathfrak m)\) be a commutative local complete Gorenstein \(k\)-algebra such that \(k\cong R/\mathfrak m\), and let \(\mathsf{MCM}(R)=\{M\in\mathsf{mod}-R|\mathrm {Ext}^i_R(M,R)=0\mathrm{ for all }i>0\}\) be the full subcategory of \textit{maximal Cohen-Macaulay} \(R\)-modules. Let \(M_0=R,\;M_1,\dots,M_t\) be pairwise non-isomorphic indecomposable \(\mathrm{MCM}\;R\)-modules, put \(M=\bigoplus_{i=1}^tM_i\), and let \(A=\mathrm {End}_R(M)\). If \(\mathrm {gldim}(A)<\infty\), \(A\) is a \textit{noncommutative resolution} (NCR) of \(R\). A particular case influencing the article, is the case where \(R\) has a finite number of indecomposable MCMs and \(M\) is their sum. Then \(\mathrm {End}_R(M)\) is the \textit{Auslander algebra} \(\mathrm {Aus}((\mathsf{MCM})\) which is a NCR.
There is a fully faithful triangle functor \(K^b(\mathrm{proj}-R)\rightarrow D^b(\mathsf{mod}-A)\) whose essential image is \(\mathsf{thick}(eA)\subseteq D^b(\mathsf{mod}-A)\), where \(e\in A\) is the idempotent corresponding to the projection on \(R\).
The \textit{classical singularity category} is defined as \(D_{sg}(R)=D^b(\mathrm{mod}-R)/K^b(\mathrm{proj}-R)\), and the authors define the \textit{relative singularity category} as the Verdier quotient category \(\Delta_R(A)=\frac{D^b(\mathrm{mod}-A)}{K^b(\mathrm{proj}-R)}\cong\frac{D^b(\mathrm{mod}-A)}{\mathrm{thick}(A)}.\) The main result in the present article relates these two categories:
``Theorem. Let \(R\) and \(R^\prime\) be MCM-representation finite complete Gorenstein \(k\)-algebras with Auslander algebras \(A=\mathrm{Aus}(\mathrm{MCM}(R))\) and \(A^\prime=\mathrm{Aus}(\mathrm{MCM}(R^\prime))\), respectively. Then the following statements are equivalent:
(i) There is an equivalence \(\underline{\mathrm{MCM}}(R)\cong\underline{\mathrm{MCM}}(R^\prime)\) of triangulated categories.
(ii) There is an equivalence \(\Delta_R(A)\cong\Delta_{R^\prime}(A^\prime)\) of triangulated categories.
The implication \((ii)\Rightarrow (i)\) holds more generally for non-commutative resolutions \(A\) and \(A^\prime\) of arbitrary isolated Gorenstein singularities \(R\) and \(R^\prime\) respectively.''
The authors use Knörrer's periodicity theorem to give nontrivial examples for (i) in the theorem.
The result is proved in a differential algebra framework. To every Hom-finte idempotent complete algebraic triangulated category \(\mathcal T\) with finitely many indecomposable objects satisfying conditions which holds for \(\mathcal T=\underline{\mathsf{MCM}}(R)\), there is an associated \textit{dg Auslander algebra} \(\Lambda_{dg}(\mathcal T)\), completely determined by \(\mathcal T\).
A recollement of categories is a collection of additive functors with certain properties, and recollements generated by idempotents, Koszul duality and the fractional Calabi-Yau property is used to prove the existence of an equivalence of triangulated categories \(\Delta_R(\mathrm{Aus}(\mathrm{MCM}(R)))\cong\mathrm{per}(\Lambda_{dg}(\underline{\mathrm{MCM}}(R))).\) This equivalence and its like, is then exploited to prove the theorem.
The article contains an appendix with a complete list of the graded quivers determining the dg Auslander algebras for ADE-singularities in all Krull dimensions.
The article shows a relation to generalized cluster categories and stable categories of special Cohen-Macaulay modules over complete rational singularities.
The article contains the necessary development of relative singularity categories, an introduction to derived categories, dg algebras and Koszul duality, complete path algebras and minimal resolutions. A nice section on the fractional Calabi-Yau property is given.
All in all, this is a very nice application of noncommutative algebraic geometry, and illustrates how the theory can, or even should, be used to obtain commutative results. isolated singularity; Gorenstein algebra; Gorenstein singularity; non-commutative resolution; singularity category; relative singularity category; perfect sheaves; dg Auslander algebra; ADE singularities; Dynkin diagrams; recollement; complete path algebra M. Kalck and D. Yang, 'Relative singularity categories I: Auslander resolutions', \textit{Adv. Math.}301 (2016) 973-1021. Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Derived categories, triangulated categories Relative singularity categories. I: Auslander 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 A new conjecture due to John McKay claims that there exists a link between (1) the conjugacy classes of the Monster sporadic group and its offspring, and (2) the Picard groups of bases in certain elliptically fibered Calabi-Yau threefolds. These Calabi-Yau spaces arise as F-theory duals of point-like instantons on ADE type quotient singularities. We believe that this conjecture, may it be true or false, connects the Monster with a fascinating area of mathematical physics which is yet to be fully explored and exploited by mathematicians. This article aims to clarify the statement of McKay's conjecture and to embed it into the mathematical context of heterotic/F-theory string-string dualities. conjugacy classes of the Monster sporadic group; Picard groups; elliptically fibered Calabi-Yau threefolds; F-theory; McKay's conjecture String and superstring theories; other extended objects (e.g., branes) in quantum field theory, \(3\)-folds, Simple groups: sporadic groups, Anomalies in quantum field theory Friendly giant meets pointlike instantons? On a new conjecture by John McKay | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities For Gorenstein quotient spaces, a direct generalization of the classical McKay correspondence in dimensions \(d\geq 4\) would primarily demand the existence of projective, crepant desingularization. Since this turned out to be not always possible, Reid asked about special classes of such quotient spaces that would satisfy the above property. In this paper, the authors prove that the underlying spaces of all Gorenstein abelian quotient singularities, which are embeddable as complete intersections of hypersurfaces in an affine space, have torus-equivariant projective crepant resolutions in all dimensions.
Conjecture: For all finite subgroups \(G\) of \(\text{SL}(d,\mathbb{C})\), for which \(\mathbb{C}^d/G\) is minimally embeddable as complete intersection (in an affine space \(\mathbb{C}^r\), \(r\geq d+1)\), the quotient space \(\mathbb{C}^d/G\) admits crepant, projective desingularizations for all \(d\geq 2\).
Main theorem. The above conjecture is true for all abelian finite groups \(G\subset SL(d,\mathbb{C})\) (for which \(\mathbb{C}^d/G\) is a complete intersection). complete intersection; McKay correspondence; crepant desingularization; Gorenstein abelian quotient singularities Dimitrios I. Dais, Martin Henk, and Günter M. Ziegler, All abelian quotient C.I.-singularities admit projective crepant resolutions in all dimensions, Adv. Math. 139 (1998), no. 2, 194 -- 239. Global theory and resolution of singularities (algebro-geometric aspects), Complete intersections, Modifications; resolution of singularities (complex-analytic aspects), Homogeneous spaces and generalizations, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) All abelian quotient C. I. -singularities admit projective crepant resolutions in all dimensions | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities For a finite subgroup \(\Gamma\subset\mathrm{SL}(2,\mathbb{C})\), we identify fine moduli spaces of certain cornered quiver algebras, defined in earlier work, with orbifold Quot schemes for the Kleinian orbifold \(\big[ \mathbb{C}^2\!/\Gamma\big]\). We also describe the reduced schemes underlying these Quot schemes as Nakajima quiver varieties for the framed McKay quiver of \(\Gamma\), taken at specific non-generic stability parameters. These schemes are therefore irreducible, normal and admit symplectic resolutions. Our results generalise our work [\textit{A. Craw} et al., Algebr. Geom. 8, No. 6, 680--704 (2021; Zbl 1494.16013)] on the Hilbert scheme of points on \(\mathbb{C}^2/\Gamma\); we present arguments that completely bypass the ADE classification. quot scheme; quiver variety; Kleinian orbifold; preprojective algebra; cornering Representations of quivers and partially ordered sets, Actions of groups on commutative rings; invariant theory, Algebraic moduli problems, moduli of vector bundles, McKay correspondence Quot schemes for Kleinian 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 Let D be a bounded symmetric domain and \(\Gamma\) \(\subset Aut(D)\) a discrete group of arithmetical type. Then the quotient space D/\(\Gamma\) is naturally endowed with the structure of a normal complex analytic space which, in general, may not be compact. Thus one encounters the problem of compactifying D/\(\Gamma\) suitably. An important standard example is provided by the moduli space \({\mathcal A}_ g\) of principally polarized abelian varieties of dimension g, a model of which is given by the quotient of the Siegel upper-half space \({\mathfrak S}_ g\) of degree g, modulo the action of the Siegel modular group Sp(2g,\({\mathbb{Z}})\). Other examples are obtained by taking the principal congruence subgroups of level \(n\geq 3\), instead of Sp(2g,\({\mathbb{Z}})\) itself, or any arithmetical subgroup of Sp(2g,\({\mathbb{R}})\) commensurable with Sp(2g,\({\mathbb{Z}})\). For these standard examples, the first compactification of the quotient spaces has been constructed by I. Satake (1956). Somewhat later, in 1966, W. Baily and A. Borel have generally shown that quotient spaces D/\(\Gamma\) of bounded symmetric domains modulo arithmetical automorphism groups can be compactified by a projective variety \(\overline{D/\Gamma}\) containing D/\(\Gamma\) as a Zariski-open subset. However, J.-I. Igusa and others have observed that the singularities of the Baily-Borel compactification \(\overline{D/\Gamma}\) are extremely complicated, which presents a serious obstacle to using algebraic geometry on \(\overline{D/\Gamma}\) in order to investigate that variety. At the same time, J.-I. Igusa (1966) gave an explicit resolution of the singularities of \(\overline{D/\Gamma}\) in the cases where \(D={\mathfrak S}_ 2\) or \(D={\mathfrak S}_ 3\), and \(\Gamma\) is commensurable with the Siegel modular group. His construction, in those particular cases, is called the Igusa compactification of the corresponding Siegel spaces. Finally, D. Mumford and his collaborators [cf. \textit{A. Ash}, \textit{D. Mumford}, \textit{M. Rapoport}, and \textit{Y.-S.Tai}, Smooth compactification of locally symmetric varieties (1975; Zbl 0334.14007)] have proven the existence of smooth compactifications for general quotient spaces D/\(\Gamma\). Their approach is based upon the theory of toroidal embeddings and, just because of its great generality, not at all easy to apply in concrete cases.
In the present paper, the authors study the three different types of compactifications (Satake, Baily-Borel, Igusa-Mumford-et al.) in one of the simplest non-trivial cases, namely for the Siegel spaces \({\mathfrak S}_ 2/\Gamma\) of degree two, where \(\Gamma\) is either Sp(2g,\({\mathbb{Z}})\) or a congruence subgroup of level \(n\geq 3\). In the first section, they reconstruct these compactifications for \({\mathfrak S}_ 2/\Gamma\) by a unified principle which is essentially based upon the common combinatorial design of a certain Tits building. This is very enlightening and reveals the basic relationship between the various compactifications from the topological and analytical viewpoint. The smooth Igusa compactification, which is treated in particularly great detail in this context, provides a nice illustration of the general toroidal compactification theory. The following sections of the paper are devoted to the study of the boundary of the Siegel space \({\mathfrak S}_ 2/\Gamma\) in its Igusa compactification (\({\mathfrak S}_ 2/\Gamma)^*\). Each irreducible component of the boundary is an elliptic modular surface (of leven n), and the further results of the authors concern the topology, the Hodge structure, and the line bundle theory of those boundary elliptic modular surfaces. More precisely, the authors compute the integral cohomology, the Chern numbers, the Todd genus, the signature, and the Hodge numbers of those surfaces. In the concluding section, various Chern classes and Chern numbers of particular line bundles (e.g., the normal bundle with respect to (\({\mathfrak S}_ 2/\Gamma)^*\), the restriction of the normal bundle to curves, the line bundle corresponding to modular forms of weight one, etc.) are computed. This is done by using results of \textit{T. Yamazaki} [Am. Math. J. 98, 39- 53 (1976; Zbl 0345.10014)] and needed, as the authors carefully explain in the introduction to their paper, as a basic tool for generalizing E. Hecke's theory of cuspidal representations of the group SL(2,\({\mathbb{Z}})\) to the groups Sp(4,\({\mathbb{Z}}/p{\mathbb{Z}})\) [cf. the authors, Proc. Natl. Acad. Sci. USA 79, 7955-7957 (1982; Zbl 0511.10022)]. Hodge theory; cohomology groups; toroidal embeddings; Siegel spaces; compactifications; modular surfaces; Chern classes; modular forms Ronnie Lee and Steven H. Weintraub, Topology of the Siegel spaces of degree two and their compactifications, Proceedings of the 1986 topology conference (Lafayette, La., 1986), 1986, pp. 115 -- 175. Compactification of analytic spaces, Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects), Complex-analytic moduli problems, Analytic theory of abelian varieties; abelian integrals and differentials, Automorphic functions in symmetric domains, Theta series; Weil representation; theta correspondences Topology of the Siegel spaces of degree two and their compactifications | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 every small resolution \(\pi:Y\to X\) of a Gorenstein threefold singularity with irreducible exceptional curve \(C\), there exists a second small resolution \(\pi^ +:Y^ +\to X\); the birational map \(Y \to Y^ +\) is a simple flop. The general hyperplane section of \(X\) is a rational double point. Every small resolution \(\pi:Y\to X\) can be constructed from a simultaneous resolution of a one-parameter deformation of a rational double point; this singularity is in general not isomorphic to the general hyperplane section. This paper shows that only \(A_ 1\), \(D_ 4\), \(E_ 6\), \(E_ 7\) and \(E_ 8\) occur as general hyperplane section; the type is determined by the multiplicity of the maximal ideal along the exceptional curve.
To prove their result, the authors use an explicit description of the simultaneous resolution of the versal deformation of the \(A\)-\(D\)-\(E\)- singularities. They compute and manipulate with the invariants of the corresponding Weyl groups. For \(E_ 8\) the resulting formula's are not given, because they are too long, but the algorithms are described (the computations were done in MAPLE and REDUCE). For \(E_ 7\) some unusual coefficients are used, to make the results comparable to those of \textit{C. C. Bramble} [Am. J. Math. 40, 351-365 (1919)]. small resolution of Gorenstein threefold singularity Katz-D, S.: Morrison, Gorenstein threefold singularities with small resolutions. J. Algebraic Geom. 1, 449--530 (1992) \(3\)-folds, Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Complex surface and hypersurface singularities Gorenstein threefold singularities with small resolutions via invariant theory for Weyl groups. Appendix 0: A good generating set in the case of \(E 8\). Appendix 1: Standard coordinates of \(E 6\). Appendix 2: Standard coordinates of \(E 7\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(A\) be the Jacobian algebra obtained from the quiver \(Q\) defined by the vertices and oriented edges of a dimer model. If the dimer model satisfies a consistency condition, then \(A\) is a non-commutative crepant resolution of its centre \(Z(A)\), which is the coordinate ring of a 3-dimensional toric Gorenstein singularity. Let \(X=\text{Spec } Z(A)\). Every projective crepant resolution of the singularity \(X\) is obtained as a fine moduli space of stable representations of \(A\) with dimension vector \((1, \dots , 1)\) denoted by \(Y\). For a certain choice of stability parameters depending on a choice of distinguished vertex \(0\in Q\), the dual of the tautological bundle on \(Y\) defines an equivalence of derived categories \(\Psi:D^b(\text{mod-}A)\to D^b(\text{coh}(Y ))\). In the special case, where the dimer model tiles the torus with triangles, then \(A\) is the skew group algebra for a finite abelian subgroup \(G \subset \mathrm{SL}(3, \mathbb{C})\), and it can be arranged \(Y\) to be the \(G\)-Hilbert scheme and the equivalence above to coincide with the derived equivalence of Bridgeland-King-Reid from the McKay correspondence.
Let \(S_i\) denote the simple \(A\)-module corresponding to vertex \(i\) in \(Q\). The main result of the paper under review proves that for any \(i\neq 0\), the object \(\Psi(S_i)\) is quasi-isomorphic to a shift of a coherent sheaf, and the derived dual of \(\Psi(S_0)\) is quasi-isomorphic to the shift by 3 of the push-forward of the structure sheaf of the fiber of \(Y\to X\) over the unique torus-invariant point. In particular, \(\Psi(S_0)\) is a pure sheaf if and only if the fiber is equidimensional. One ingredient of the proof involves establishing a link between the objects \(\Psi(S_i)\) that have non-vanishing cohomology in degree zero and certain walls of the GIT chamber containing the stability parameter. This result in combination with other known results provide the Geometric Reid's recipe which is the dimer model analogue of the Geometric McKay correspondence in dimension three proven by Logvinenko. Geometric Reid's recipe provides a description of the objects \(\Psi(S_i)\). dimer models; crepant resolution; quiver representation Bocklandt, R; Craw, A; Quintero Vélez, A, Geometric reid's recipe for dimer models, Math. Ann., 361, 689-723, (2015) McKay correspondence, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Global theory and resolution of singularities (algebro-geometric aspects), Geometric invariant theory, Derived categories and associative algebras, Representations of quivers and partially ordered sets Geometric Reid's recipe for dimer models | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We intend to outline some new developments in deformation theory of two-dimensional quotient singularities. The various topics, together with short summaries, are the following:
1. \textbf{Quotients of complex analytic spaces and quotient surface singularities}.
We discuss the concept of quotient singularities in a general context. Then we outline the classification in dimension 2. Quotients of \(\mathbb{C}^2\) by cyclic subgroups of \(\text{GL}(2,\mathbb{C})\) (cyclic quotients) and their equations are discussed.
2. \textbf{Resolutions of singularities.}
By way of the concept of resolutions we introduce the notion of rational singularities. We prove that quotient surface singularities are rational and describe the structure of their resolutions.
3. \textbf{Deformations of singularities.} We explain the concept of a deformation. For isolated singularities, there exists a semi-universal deformation (without proof). First examples for (semi-universal) deformations are given.
4. \textbf{Simultaneous resolutions.}
For rational singularities, the importance of the Artin-component (of simultaneously resolvable deformations after finite base change) will be recognized. Further we discuss among other topics the cotangent cohomology and recent results on the vector space of first order deformations in various examples.
5. \textbf{Threefolds and deformations of surface singularities.}
After a short excursion into the theory of three-dimensional canonical singularities, we shall state the result of Kollár and Shepherd-Barron which establishes a bijective correspondence between the non-embedded irreducible components of the base space of the semi-universal deformation of a quotient surface singularity and a certain set of partial resolutions \((P\)-resolutions) of such singularities. As an application, we illustrate the structure of components of the base space of cyclic quotient surface singularities.
6. \textbf{M-resolutions and deformations of quotient singularities.}
The general fibre over a non-embedded irreducible component of the base space of a quotient surface singularity is smooth. By a certain modification of a \(P\)-resolution, we come to a so-called \(M\)-resolution, which allows us to compute over each component the monodromy group. deformation; two-dimensional quotient singularities K. Behnke and O. Riemenschneider, Quotient surface singularities and their deformations, Singularity theory (Trieste 1991), World Scientific, Singapore (1995), 1-54. Deformations of complex singularities; vanishing cycles, Singularities of surfaces or higher-dimensional varieties, Complex surface and hypersurface singularities Quotient surface singularities and their deformations | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The article under review is an expository one and discusses two basic cases of resolution of singularities. It starts with the existence of an equivariant resolution of singularities for a toric variety \(X_\Delta\) [\textit{G. Kempf, F. Knudsen, D. Mumford} and \textit{B. Saint-Donat}, Toroidal embeddings I, Lect. Notes in Mathematics 339. Berlin-Heidelberg-New York, Springer-Verlag (1973; Zbl 0271.14017)], proved by constructing a regular subdivision of the fan \(\Delta\). Next, a result of Khovansky [\textit{A. Varchenco}, Invent. Math. 37, 253--262 (1976; Zbl 0333.14007)] gives the existence of a weak resolution of singularities for a hypersurface \(H\subset\mathbb{C}^{n+1}\) defined by a non-degenerate irreducible polynomial \(f\). Its proof is based on subdividing the dual fan of the Newton polyhedron \(\Gamma_+(f)\) to obtain a regular subdivision of the cone \({\mathbb{R}}_{\geq 0}^{n+1}\). After introducing briefly canonical divisors and proving the adjunction formula, canonical, terminal, log-canonical and log-terminal singularities are defined. The last theorem gives a necessary and sufficient condition for a non-degenerate hypersurface \(H\) to have canonical or log-canonical singularities at 0 in terms of its Newton polyhedron \(\Gamma_+(f)\). Though the author works over the field of complex numbers, all results remain valid for an arbitrary algebraically closed field. Newton polygone; resolution of singularities; toric variety Ishii, S.; Brasselet, J-P (ed.); etal., A resolution of singularities of a toric variety and non-degenerate hypersurface, 354-369, (2007), Hackensack Global theory and resolution of singularities (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies, Hypersurfaces and algebraic geometry A resolution of singularities of a toric variety and a non-degenerate hypersurface | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \((X,C)\) be a pair where \(X\) is a germ of a normal two-dimensional singularity and \(C \subset X\) is a germ of smooth curve. Jaffe gave a complete description of such pairs up to isomorphisms when \(X\) is a rational double point in terms of the associated Dynkin diagram. In this paper that result is extended to a more general class of rational singularities. The description of the set of isomorphism classes of \((X,C)\) is in terms of the points that the fundamental cycle \(E\) in the minimal resolution of the singularity cuts out on the strict transform of \(C\) (via the resolution map). The result relies on some properties pointed out in a thorough study of the minimal resolution of the singularity. permissible point; Dynkin diagram; rational singularities; minimal resolution M. Manetti, On smooth curves passing through rational surface singularities. Math. Z.218, 375--386 (1995). Singularities of surfaces or higher-dimensional varieties, Curves in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) On smooth curves passing through rational surface singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For part I see Singularities, Summer Inst., Arcata/Calif. 1981, Proc. Symp. Pure Math. 40, Part 2, 675-680 (1983; Zbl 0545.55005); see also Invent. Math. 70, 169-218 (1982; Zbl 0508.20020).]
Let \(\Gamma\) be an arithmetic subgroup of an algebraic group G over \({\mathbb{Q}}\) and consider the Baily-Borel-Satake compactification of the quotient \(\Gamma\setminus X\), where X is the symmetric space associated to G and assumed to be Hermitian. Then the conjecture is that the middle perversity intersection homology of this compactification is the \(L_ 2\)-cohomology of \(\Gamma\setminus X\). The conjecture is verified here for some low rank \((r=2,3)\) examples for G: Sp(4), SU(p,2), Sp(6). This is done by checking a characterization of intersection homology and to this end laboriously computing the \(L_ 2\)-cohomology of (a suitable system of) neighbourhoods on the singular strata \(S^{j(1)},...,S^{j(r)}\quad (j(i)\) denoting codimension) of the completion, applying a Künneth formula.
In an appendix it is shown that the half-sum of the positive roots of G (assumed to be of type \(B_ r\), with the simple roots \(\beta_ 1=\epsilon_ 1-\epsilon_ 2,...,\beta_{r-1}=\epsilon_{r-1}- \epsilon_ r\), \(\beta_ r=\epsilon_ r)\), restricted to a maximal \({\mathbb{Q}}\)-split torus, is \(\sum j(i)\beta_ i.\) The paper has sections on compactification, \(L_ 2\)-cohomology and intersection cohomology. Baily-Borel-Satake compactification; middle perversity intersection homology; \(L_ 2\)-cohomology Steven Zucker, \?\(_{2}\)-cohomology and intersection homology of locally symmetric varieties. II, Compositio Math. 59 (1986), no. 3, 339 -- 398. (Co)homology theory in algebraic geometry, Cohomology theory for linear algebraic groups, Products and intersections in homology and cohomology, Homogeneous spaces and generalizations \(L_ 2\)-cohomology and intersection homology of locally symmetric varieties. II | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(M_0\) be a smooth complex threefold with trivial canonical bundle \(\omega_{M_0}\) acted on by a finite group \(G\) of automorphisms acting trivially on \(\omega_{M_0}\). A conjecture of Dixon-Harvey-Vafa-Witten says that there always exists a desingularization \(M_0/G\) with trivial canonical bundle, and predicts its Euler number. This conjecture has a local form from which it follows: If \(G\) is a finite subgroup of \(SL(3,{\mathbb{C}})\), there exists a crepant (i.e. with trivial canonical bundle) desingularization of \({\mathbb{C}}^3/G\), and its Euler number is the number of conjugacy classes of \(G\). The conjecture is now completely solved: There is a list of all finite subgroups of \(SL(3,{\mathbb{C}})\) due to Miller, Blichfeldt and Dickson, and a case-by-case analysis, of which this paper is part, shows that there always exists a crepant resolution, constructed by an explicit sequence of blow-ups. The formula for the Euler number follows from these explicit constructions; it also has an independent general proof by the generalization of the MacKay correspondence to dimension \(3\) by \textit{Y. Ito} and \textit{M. Reid}, in: Higher-dimensional complex varieties, Proc. Int. Conf., Trento 1994, 221-240 (1996; Zbl 0894.14024). The construction of a crepant resolution of \({\mathbb{C}}^3/H_{168}\) is pretty straightforward: One makes \(4\) successive blow-ups of singular curves until all the singular points disappear. The fact that the resolution obtained in this way is crepant follows from a remark of Reid (1979). Calabi-Yau threefolds; McKay correspondence; crepant resolutions; complex threefold; finite group of automorphisms; Euler number G. Markushevich, ''Resolution of \(\mathbb{C}\)3/H 168,''Math. Ann.,308, 279--289 (1997). \(3\)-folds, Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Group actions on varieties or schemes (quotients), Birational automorphisms, Cremona group and generalizations Resolution of \(\mathbb{C}^3/H_{168}\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This article is an important study of homological mirror symmetry for hypersurface cusp singularities. A cusp singularity \((\bar{Y}, p)\) is the germ of an isolated, normal surface singularity such that the exceptional divisor of a minimal resolution \(\pi: Y \to \bar{Y}\) is a cycle of smooth rational curves intersecting transversely. Denote this cycle of \(\mathbb{P}^1\)'s by \(D\). Note that the germ of a cusp singularity \((Y,p)\) is uniquely determined by the self-intersection numbers of the components of the associated cycle \(\pi^{-1}(p)=D\).
Cusp singularities naturally come in dual pairs. That is, given a cusp singularity \((\bar{Y}, p)\) with associated cycle \(D\), there is a natural dual cusp singularity \((\bar{Y}', p')\) with associated cycle \(D'\). In [Ann. Math. (2) 114, 267--322 (1981; Zbl 0509.14035)], \textit{E. Looijenga} showed that if the cusp singularity with associated cycle \(D'\) is smoothable, then there exists a smooth rational surface \(Y\) with anti-canonical divisor \(D\), whose components have the same self intersections as the cycle associated to the dual cusp singularity. Moreover, the Looijenga conjecture asserts that if there exists a smooth surface \(Y\) and an anti-canonical cycle \(D\), then the cusp singularity with associated cycle \(D'\) is smoothable. This conjecture was proven in the context of mirror symmetry in [\textit{M. Gross} et al., Publ. Math., Inst. Hautes Étud. Sci. 122, 65--168 (2015; Zbl 1351.14024)].
A hypersurface cusp singularity is a singularity given by a single polynomial equation
\[
T_{p,q,r}(x, y,z) = x^p + y^q + z^r + axyz
\]
where \(a\) is a non-zero complex number, and \((p, q,r)\) is a triple of positive integers satisfying
\[
\frac{1}{p}+\frac{1}{q}+\frac{1}{r} \leq 1
\]
A smoothing of a hypersurface cusp singularity \(T_{p,q,r}(x, y,z)\) is given by its Milnor fiber \(\tau_{p,q,r}\).
In the paper, the Milnor fiber is exhibited as a Lefschetz fibration \(\Xi: \tau_{p,q,r} \to \mathbb{C}\), with smooth fibre \(M\), and the following equivalences of categories are proven.
\[
D^b\operatorname{Fuk}^{\to}(\Xi) \simeq D^b\operatorname{Coh}(Y_{p,q,r})
\]
where \(\operatorname{Fuk}^{\to}(\Xi)\) stands for the Fukaya-Seidel category of \(\Xi\).
\[
D^\pi \operatorname{Fuk}(M) \simeq \operatorname{Perf}(D)
\]
where \(D^\pi \operatorname{Fuk}(M)\) denotes the derived split closure of the Fukaya category of the fibre \(M\) and \(\operatorname{Perf}(D)\) is the category of perfect complexes of algebraic vector bundles on \(D\).
\[
D^bW(\tau_{p,q,r}) \simeq D^b \operatorname{Coh}(Y_{p,q,r} \setminus D)
\]
where \(W(T_{p,q,r})\) is the wrapped Fukaya category of \(\tau_{p,q,r}\). homological mirror symmetry; cusp singularities Keating, A.: Homological mirror symmetry for hypersurface cusp singularities. Preprint (2015), arXiv:1510.08911 Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category, Mirror symmetry (algebro-geometric aspects) Homological mirror symmetry for hypersurface cusp singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This article considers the relation between deformations of a rational surface singularity with a reflexive module, and deformations of a partial resolution of the singularity with the locally free strict transform of the module. The results indicate how a family of small resolutions of a 3-dimensional index one terminal singularity and its flop are obtained by blowing up in a maximal Cohen-Macaulay module and its syzygy.
The article is motivated by the work on the geometrical McKay correspondence which can be said to give a one-to-one correspondence between the isomorphism classes of indecomposable reflexive modules \(\{M_i\}\) and the prime components \(\{E_j\}\) of the exceptional divisor in the minimal resolution \(\tilde X\rightarrow X\) of a rational double point (RDP), that is \(A_n\), \(D_n\), \(E_{6 - 8}.\)
For a natural class of special reflexive modules named Wunram modules after its inventor, the correspondence holds for any rational surface singularity. \textit{M. Van den Bergh} [Duke Math. J. 122, No. 3, 423--455 (2004; Zbl 1074.14013)] used the endomorphism ring of a higher dimensional Wunram module to prove derived equivalences for flops, and this again led to attention to the \(2\)-dimensional case with interesting results by \textit{O. Iyama} and \textit{M. Wemyss} [Math. Z. 265, No. 1, 41--83 (2010; Zbl 1192.13012); Ill. J. Math. 55, No. 1, 325--341 (2011; Zbl 1258.13015)] and \textit{M. Wemyss} [Math. Ann. 350, No. 3, 631--659 (2011; Zbl 1233.14012)].
In this article, the authors prove that blowing up a rational surface singularity \(X\) in a reflexive module \(M\) gives a partial resolution \(f:Y\rightarrow X\) where \(Y\) is normal, dominated by the minimal resolution , and where the strict transform \(\mathcal M=f^\Delta(M)\) is locally free. This partial resolution is determined by the first Chern class \(c_1(\mathcal F)\) of the strict transform \(\mathcal F\) of \(M\) to \(\tilde X.\) Thus, in particular, any partial resolution dominated by the minimal resolution is given by blowing up in a Wunram module, and the authors mention the RDP-resolution obtained by contracting the \((-2)\)-curves in the minimal resolution in particular; this is given by blowing up in the canonical module \(\omega_X.\)
Consider the category of deformations \(\text{Def}_{Y,\mathcal M}\) of the pair \((Y,\mathcal M)\) blowing down to \((X,M)\). The main result in the present article says that the blowing down map \(\alpha:\text{Def}_{Y,\mathcal M}\rightarrow\text{Def}_{X,M}\) is injective and that it commutes with the forgetful map \(\beta:\text{Def}_{Y,\mathcal M}\rightarrow\text{Def}_{Y}\) and the blowing down map \(\delta:\text{Def}_Y\rightarrow\text{Def}_X.\) Furthermore, the forgetful map \(\beta\) is smooth and an isomorphism in many situations. The blowing down map \(\delta\) is a Galois covering onto the Artin component \(A\) on spaces, which for RDPs equals \(\text{Def}_X\). In general, it not injective, making the injectivity of \(\alpha\) surprising. The authors prove that \(\beta\) is an isomorphism if \(M\) is Wunram, implying that \(\delta\) factors through a closed embedding \(\alpha\beta^{-1}:\text{Def}_Y\subseteq\text{Def}_{(X,M)}\) realizing deformations of the pair as conjectured by \textit{C. Curto} and \textit{D. R. Morrison} [J. Algebr. Geom. 22, No. 4, 599--627 (2013; Zbl 1360.14053)] in the RDP case.
A deformation of the pair \((X,M)\) in the geometric image of \(\text{Def}_{(Y,\mathcal M)}\) lifts to a deformation of \((Y,\mathcal M)\) without any base change. In general, \(\text{Def}_{X,M}\) is not dominated by \(\text{Def}_{(Y,\mathcal M)}\) even for RDPs. A main ingredient in Wahl's proof that the covering \(\text{Def}_{\tilde X}\rightarrow A\) has Galois action by a product of Weyl groups is the injectivity of \(\delta\) in the case that \(Y\) is the RDP resolution. This follows directly from the authors result because \(\text{Def}_{X,\omega_X}\cong\text{Def}_{X}\). The results indicate that there are interesting relations to \(\text{Def}_{X}\), for instance the component structure.
The main application of the result above is a generalization of three conjectures of Curto and Morrison [loc. cit.] concerning the nature of small partial resolutions of \(3\)-dimensional index one terminal singularities and their flops. When \(g:W\rightarrow Z\) is a small partial resolution and \(X\subseteq Z\) is a sufficiently generic hyperplane section with strict transform \(f:Y\rightarrow X\), a result of Reid states that \(f\) is a partial resolution of an RDP. Thus \(g\) is a \(1\)-parameter deformation of \(f\) and so an element in \(\text{Def}_Y\). The authors prove that \(Y\) then is the blowing up of \(X\) in a reflexive module \(M\). As a consequence, \(\alpha\beta^{-1}\) takes this \(g\) to a \(1\)-parameter deformation of \((Z,N)\) of the pair \((X,M).\) Verbatim:
Corollary. There is a maximal Cohen-Macaulay \(\mathcal O_Z\)-module \(N\) such that (i) The small partial resoltuion \(W\rightarrow Z\) is given by blowing up \(Z\) in \(N.\) (ii) Blowing up \(Z\) in the syzygy module \(N^+\) of \(N\) gives the unique flop \(W^+\rightarrow Z\). (iii) The length of the flop equals the rank of \(N\) if the flop is simple.
A version of this statement is given for flat families of small partial resolutions and flops. It is proved that there is a family of pairs \((\mathbf{X},\mathbf{M})\) in \(\text{Def}_{(X,M)}\) such that the blowing up of \(\mathbf{X}\) in \(\mathbf{M}\) and in the syzygy \(\mathbf{X}^+\) give two simultaneous partial resolutions \(\mathbf{Y}\rightarrow\mathbf{X}\leftarrow\mathbf{Y}^+\) inducing any local family of flops \(g\) by pullback, for any \(g\) with hyperplane section \(f\). In fact, \(\mathbf{X}\) equals \(A_1, D_4, E_6, E_7, E_8\) for \(l=1,2,3,4,5,6\) respectively, so that the result above proves that there is in each case a unique reflexive module \(M\) of rank \(l\) such that any simple flop of length \(l\) is obtained by pullback from the \(\mathbf{Y}\rightarrow\mathbf{X}\leftarrow\mathbf{Y}^+.\) This gives the universal simple flop of length \(l\) realized as blowing-ups in families of reflexive modules (as suggested by Curto and Morrison [loc. cit.]).
The RDPs are hypersurfaces, and any maximal Cohen-Macaulay module is given by a matrix factorization. The conjectures of Curto and Morrison [loc. cit.] are stated by matrix factorizations, and they are verified for \(A_n\) and \(D_n\) by brute force computations. The present argument is conceptual, coordinate-free, and makes the conjectures transparent.
The singularities considered in this work will all be henselisations of finite algebras and the results will therefore have finite type representations locally in the étale topology.
Work by Donovan, Wemyss et.al. links properties of various noncommutative algebras to flops. This involves quiver algebras, mutations, tilting theory and GIT-constructions with endomorphism algebras as input. This work offers a direct proof of the original Curto-Morrison conjectures using deformation theory where the blowing up ideal for the small, partial resolution is obtained directly from the \(2\)-dimensional Wunram module. Any flop with fixed RDP hyperplane section and Dynkin diagram is a pullback from a pair of universal blowing ups.
The article is impressing. The deformation functors are studied conceptually, more as fibred categories than as functors, and natural transformations are transformed into maps between the categories of resolutions of singularities. The article gives a lot of techniques that that can be used in the study of more general contractions, and shows a brilliant use of deformation theory in general. Also, the article is self contained with respect to the deformation theory, and contains all preliminaries needed for understanding the importance of the results. flatifying blowing-up; maximal Cohen Macaulay module; simultaneous partial resolution; small resolution; rational double point; RDP; matrix factorization; deformation of algebras; deformation of rational singularities; deformations of exceptional module; partial resolution; domination of resolution; contracting curves; strict transform; Wunram module; blowing up Deformations of singularities, Minimal model program (Mori theory, extremal rays), Stacks and moduli problems, McKay correspondence Deformations of rational surface singularities and reflexive modules with an application to 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 A very beautiful connection was found by Brieskorn and Slodowy between simple singularities (rational double points) and simple Lie algebras. In the paper under review the author seeks an analogue of this connection in the class of 1-dimensional simple singularities which are complete intersection in \({\mathbb{C}}^ 3.\) These singularities are classified by \textit{M. Giusti} [Singularities, Summer Inst., Arcata/Calif. 1987, Proc. Symp. Pure Math. 40, Part 1, 457-494 (1983; Zbl 0525.32006)] and they are labelled \(S_{\mu} (\mu =5,6,7,...)\), \(T_ 7,T_ 8,T_ 9,U_ 7,U_ 8,U_ 9,W_ 9,W_{10},Z_ 9,Z_{10}\). The author associate to \(S_{\mu}^ a \)diagram called \(D=D_ k[*]\), \(k=\mu -1\) which is constructed from the Dynkin diagram \(D_ k\) by adding one distinguished vertex. They are connected as follows: The diagram determines a torus embedding \({\mathcal X}(D)\) and the parameter space of the semi-universal deformation of the singularity is identified with \({\mathcal X}(D)/W_ 2(D)\), where \(W_ 2(D)\) is an analogue of the Weyl group, and this identification respects the discriminants of the both spaces. A similar result is proved for \(T_{\mu}\) and \(E_{\mu}[*]\) \((\mu =6,7,8)\). extended Dynkin diagram; 1-dimensional simple singularities; complete intersection; torus embedding; semi-universal deformation K. Wirthmüller , Torus embeddings and deformations of simple singularities of space curves , Acta Math. 157 (1986), 159-241. Singularities of curves, local rings, Deformations of complex singularities; vanishing cycles, Deformations of singularities, Projective techniques in algebraic geometry, Embeddings in algebraic geometry Torus embeddings and deformation of simple singularities of space 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 At its most basic, mirror symmetry is a local isomorphism called the mirror map between the complex moduli on the \(B\)-model and the Kähler moduli on the \(A\)-model. The complex moduli space parametrizes deformations of complex structure (of the Calabi-Yau manifold under consideration). The Kähler moduli parametrizes the complexified Kähler classes of the mirror.
This local isomorphism was originally defined near special points called the large complex structure limits, which can be thought of as having monodromy that is maximally unipotent. Locally, each such degeneration corresponds to the complexified Kähler cone of the mirror to the degenerating family. Conjecturally, the Kähler cones glue together to form the stringy Kähler moduli space, which should be globally isomorphic to the entire complex moduli.
On both sides, there are additional special points, such as the Geepner and conifold points. Different correspondences refer to equivalences of invariants defined for various couples of special points and the term \textit{global mirror symmetry} introduced by \textit{A. Chiodo} and \textit{Y. Ruan} [Invent. Math. 182, No. 1, 117--165 (2010; Zbl 1197.14043)] refers to having a unified picture, i.e., a global isomorphism on the entire complex and stringy Kähler moduli spaces.
The paper under review is the 4th in a series of papers by various combinations of the authors and Kravitz and Ruan, where global mirror symmetry for simple elliptic singularities is investigated.
Start with an invertible simple elliptic singularity (ISES) \(W\). Simple elliptic singularity means that \(W=0\) is the germ of a singularity such that the exceptional divisor of its minimal resolution is an elliptic curve. Such a singularity can always be given by a quasi-homogeneous non-degenerate polynomial \(W\). Invertible means that the exponent matrix of \(W\) is invertible. The classification of such ISES is given in Table 1.1 of the paper and they correspond to Dynkin diagrams of type \(E_6\), \(E_7\) and \(E_8\).
To such a \(W\), consider its miniversal deformation space \(\mathcal{S}\). This is the (local) complex moduli investigated. The authors consider special limit points \(\sigma\in\mathcal{S}\), namely those for which the corresponding deformation of \(W\) no longer has an isolated critical point at \(0\). They explain how to associate to such \(\sigma\) the Saito-Givental ancestor potential \(\mathcal{A}_W^{SG}(\sigma)\).
The global mirror symmetry expectation is introduced in Conjecture 1.3, which states that the mirror of special limit points in \(\mathcal{S}\) should be mirror to either of the two following: {
- The Fan-Jarvis-Ruan-Witten (FJRW) theory of a simple elliptic singularity.
- The Gromov-Witten (GW) theory of an elliptic orbifold \(\mathbb{P}^1\).
} The mirror relationship is described as an equality between the corresponding ancestor potentials. The authors prove that Conjecture 1.3 holds for Geepner points (Theorem 1.4) and for Fermat simple elliptic singularities (Theorem 1.5).
The paper is overall well-written and includes a good amount of background information. It is instructive to see the full picture of global mirror symmetry unfolding. mirror symmetry; global mirror symmetry; simple elliptic singularities; Saito-Givental theory; primitive forms; Gromov-Witten theory; FJRW theory; quasi-modular forms T. Milanov and Y. Shen, Global mirror symmetry for invertible simple elliptic singularities, preprint (2014), . Mirror symmetry (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Singularities in algebraic geometry, Deformations of singularities Global mirror symmetry for invertible simple elliptic singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The article is based on lectures that the author gave at the CIME Summer School ``Representation Theory and Complex Analysis'' in Venice in 2004.
The Langlands program, which was conceived as a bridge between Number Theory and Automorphic Representations, has recently expanded into many other areas, as for instance Geometry and Quantum Field Theory.
The article focuses on the geometric Langlands correspondence, which is a particular brand of the general theory. Its framework is the following: let \(X\) be a smooth projective curve over \(\mathbb C\) (the field of complex numbers) and let \(G\) be a simple complex Lie group. Let \({}^LG\) denote the Langlands dual group of \(G\). Then let \(\mathcal F\) be a holomorphic principal \({}^LG\)-bundle on \(X\) equipped with a holomorphic connection \(\Delta\). Note that the pair \(E:=(\mathcal F,\Delta)\) may also be thought of as a \({}^LG\)-local system on \(X\). Let \(\text{Bun}_G\) denote the moduli stack of holomorphic \({}^LG\)-bundles on \(X\).
The global (unramified) Langlands correspondence is supposed to assign to each pair \(E=(\mathcal F,\Delta)\) as above an object \(\mathcal{A}ut_E\) (called a Hecke eigensheaf with eigenvalue \(E\)) on \(\text{Bun}_G\).
The subject of the article is the ramified geometric Langlands correspondence, that is, the case where one allows the connection \(\Delta\) to be singular at finitely many points of \(X\) (when \(\Delta\) has no pole, the corresponding local system is called unramified). Here the moduli stack \(\text{Bun}_G\) has to be replaced by the enhanced moduli stack, which is the moduli stack of \(G\)-bundles together with the level structure at the ramification points. Then the Langlands correspondence will assign to a flat \({}^LG\)-bundle \(E=(\mathcal F,\Delta)\) with ramification at the points \(y_1\), \(\ldots\), \(y_n\) a category \(\mathcal{A}ut_E\) of Hecke eigensheaves on the corresponding enhanced moduli stack with eigenvalue \(E|_{X\backslash\{y_1,\ldots,y_n\}}\), which is a subcategory of the category of (twisted) \(\mathcal D\)-modules on this moduli stack.
The article reviews a previous work of the author, jointly with Gaitsgory, which provides an approach to construct the categories \(\mathcal{A}ut_E\) and to describe them in terms of the Langlands group \({}^LG\). The idea goes back to the construction of Beilinson and Drinfeld of the unramified Langlands correspondence, which may be interpreted in terms of a localization functor (this construction is reviewed in the first section of the article and serves as a prototype for the more general constructions). Functors of this kind were introduced by Beilinson and Bernstein in representation theory of Lie algebras. In the present situation this functor sends representations of the affine Kac-Moody algebra \(\hat {\mathfrak g}\), that is, the universal one-dimensional central extension of the formal loop algebra \({\mathfrak g}((t))\) -- here \({\mathfrak g}\) is a simple Lie algebra, with \(G\) denoting the corresponding connected, simply-connected algebraic group -- to twisted \(\mathcal D\)-modules on \(\text{Bun}_G\), or its enhanced versions. These \(\mathcal D\)-modules may be viewed as sheaves of conformal blocks naturally arising in the framework of Conformal Field Theory.
The representation categories of \(\hat {\mathfrak g}\) have a parameter \(\kappa\), called the level, an invariant bilinear form on \({\mathfrak g}\), which determines the scalar by which a generator of the one-dimensional center of \(\hat {\mathfrak g}\) acts on representations. The author considers a particular value \(\kappa_c\) of this parameter, called the critical level, which is equal to minus one half of the Killing form. The fact that the completed enveloping algebra of an affine Kac-Moody algebra acquires a particular large center at the critical level makes the structure of the corresponding category \(\hat{\mathfrak g}_{\kappa_c}\) very rich and interesting. (The affine Kac-Moody algebra \(\hat{\mathfrak g}_{\kappa}\) is defined as the central extension \(0\to \mathbb C\,{\mathbf 1}\to \hat{\mathfrak g}_{\kappa}\to {\mathfrak g}((t))\to 0\).) The author, jointly with Feigin, proved previously that this center is canonically isomorphic to the algebra of functions on the space \(\text{Op}_{{}^LG}(D^\times)\) of \({}^LG\)-opers on the formal punctured disc \(D^\times=\text{Spec}\,\mathbb C((t))\). Opers are bundles on \(D^\times\) with flat connection and an additional datum (the precise definition is recalled in the article). They were introduced by Drinfeld and Sokolov, and their connection to representations of affine Kac-Moody algebras at the critical level was discovered by the author of the article, jointly with Feigin.
For each \(\chi\in \text{Op}_{{}^LG}(D^\times)\) there is a ``fiber'' category \(\hat{\mathfrak g}_{\kappa_{c}}\)-mod\(_{\chi}\) whose objects are \(\hat{\mathfrak g}\)-modules on which the center acts via the central character corresponding to \(\chi\). Applying the localization functors to these categories and their \(K\)-equivariant subcategories \(\hat{\mathfrak g}_{\kappa_{c}}\)-mod\(_{\chi}^{K}\) for various subgroups \(K\subset G[[t]]\), one obtains categories of Hecke eigensheaves on the moduli spaces of \(G\)-bundles on \(X\) with level structures. It follows that the localization functor provides a power tool for converting local categories of representations of \(\hat{\mathfrak g}\) into global categories of Hecke eigensheaves. This new phenomenon does not seem to have any obvious analogue in the classical Langlands correspondence. geometric Langlands correspondence; affine Kac-Moody algebras; opers Frenkel, Edward, Ramifications of the geometric {L}anglands program, Representation Theory and Complex Analysis, Lecture Notes in Math., 1931, 51-135, (2008), Springer, Berlin Geometric Langlands program: representation-theoretic aspects, Geometric Langlands program (algebro-geometric aspects), Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras Ramification 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 representation theory of commutative algebras is based on Auslander-Reiten (AR) duality for isolated singularities. This article establishes a version of AR duality for singularities with one-dimensional singular loci. Then the stable categories of Cohen Macaulay (CM) modules over Gorenstein singularities with one dimensional singular loci have a generalized Calabi-Yau property, meaning that it is possible to apply the methods of cluster tilting (CT) in representation theory to study singularities. Of particular interest is the AR theory of simple surface singularities which have only finitely many indecomposable CM modules, and the AR algebras \(\text{End}(\oplus_{M\text{ ind. CM}}M)\) are polite. This is not the case for singularities of dimension greater than two, then the representations are not finite, and the AR-algebras are not nice. To obtain the correct category on which higher AR theory is good, \textit{O. Iyama} [Adv. Math. 210, No. 1, 22--50 (2007; Zbl 1115.16005)] introduced a notion of a maximal \(n\)-orthogonal subcategory and maximal \(n\)-orthogonal module for the category mod \(\Lambda\), called CT subcategories and CT modules. In the case of a higher dimensional CM singularity \(R\), the definition of a maximal \(n\)-othogonal subcategory and modules is applied to CM \(R\) and it is hoped that this handles higher-dimensional geometric problems. Strong evidence is given when \(R\) is a three dimensional normal isolated Gorenstein singularity: Then it is known that such objects have a relationship with noncommutative crepant resolutions (NCCR) given by \textit{M. Van den Bergh} [Duke Math. J. 122, No. 3, 423--455 (2004; Zbl 1074.14013)].
The study if maximal \(n\)-orthogonal modules in CM \(R\) is not suited for studying non-isolated singularities because the Ext vanishing condition is too strong. So one needs a subcategory of CM \(R\) that can take the place of the maximal \(n\)-orthogonal subcategories. This is the main goal this article, but it gives more: The authors develop a theory which can deal with singularities in the crepant partial resolutions. As the endomorphism rings of maximal orthogonal modules have finite global dimension, these cannot work for the purpose. The notion of maximal modifying modules are introduced as corresponding to shadows of maximal crepant partial resolutions. This level always exist geometrically, but it can happen to be smooth. The level of generality is thus to view the case when the geometry is smooth as a coincidence: everything that can be done with NCCRs should be possible to do with maximal modifying modules.
When curves are flopped between varieties with canonical singularities which are not terminal, this doesn't take place on the maximal level. Anyway, it should be homologically understandable. That is the motivation for defining modifying modules. This also leads to the development of the theory of mutation for modifying modules.
The authors remark that some parts of the theory of (maximal) modifying modules are analogues of cluster tilting theory, particularly for rings of Krull dimension three. Some of the main properties of CT theory are still true in the present setting, but new features also appear, e.g. a mutation sometimes does not change the given modifying module, which is a necessary feature since it reflects the geometry of partial crepant resolutions.
The present theory depends on a more general noncommutative setting. The authors use the language of singular Calabi-Yau algebras: Let \(\Lambda\) be an \(R\)-algebra, finitely generated over \(R\) (module finite). Then for \(d\in\mathbb Z\), \(\Lambda\) is called \(d\)-Calabi-Yau (=d-CY) if there is an isomorphism
\(\text{Hom}_{D(\text{Mod}\Lambda)}(X,Y[d])\simeq D_0\text{Hom}_{D(\text{Mod}\Lambda)}(Y,X)\) for all \(X\in D^{b}(\text{fl}\Lambda)\), \(Y\in D^b(\text{mod}\Lambda)\) where \(D_0\) is the Matlis-dual. \(\Lambda\) is called singular Calabi-Yau (d-sCY) if the isomorphism holds for all \(X\in D^{b}(\text{fl}\Lambda)\), \(Y\in K^b(\text{proj}\Lambda)\)
When \(\Lambda=R\), it is known that \(R\) is \(d\)-sCY iff \(R\) is Gorenstein and equi-codimensional with dim \(R=d\). Notice that purely commutative statements are proved in a commutative setting.
Now, let \(R\) be an equi-codimensional CM ring of dimension \(d\) with canonical module \(\omega_R\). CM \(R\) is the category of CM \(R\)-modules, \(\underline{\text{CM}} R\) is the stable category, and \(\overline{CM}R\) is the costable category. The AR stranslation is \(\tau=\text{Hom}_R(\Omega^d\text{Tr}(-),\omega_R):\underline{\text{CM}} R\rightarrow\overline{CM}R\).
When \(R\) is an isolated singularity, one of the fundamental properties of the category CM \(R\) is the existence of AR duality: \(\underline{\text{Hom}}(X,Y)\equiv D_0\text{Ext}^1_R(Y,\tau X)\) for all \(X,Y\in\) CM \(R\). Letting \(D_1:=\text{Ext}^{d-1}_R(-,\omega_R)\) be the duality on the category of CM-modules of dimension 1, the authors prove that AR duality generalizes to mildly non-isolated singularities, in a precise sense. This generalised AR-duality also implies a generalized \((d-1)\)-Calabi-Yau property of the triangulated category \(\underline{\text{CM}} R\), and the symmetry in the Hom groups gives the tool needed to move from the CT level to the MM level. It is analogues to the symmetry given in cluster theory.
Recall that if \(R\) is a CM ring and \(\Lambda\) a module finite \(R\)-algebra, then \(\Lambda\) is an \(R\)-order if \(\Lambda\in\text{CM}R\), an \(R\)-order \(\Lambda\) is non singular if gl.dim\(\Lambda_{\mathfrak p}=\text{dim}R_{\mathfrak p}\) for all primes \(\mathfrak p\subset R\), and an \(R\)-order \(\Lambda\) has isolated singularities if \(\Lambda\) is a non-singular \(R_{\mathfrak p}\)-order for all non-maximal primes \(\mathfrak p\subset R\). Then for \(R\) CM, a noncommutative crepant resolution (NCCR) of \(R\) is \(\Gamma:=\text{End}_R(M)\) where \(M\) is reflexive and non-zero such that \(\Gamma\) is a non-singular \(R\)-order.
The article gives the following classification of cluster tilting (CT) modules: For a field of characteristic zero, \(S=k[x_1,\dots,x_d]\), and \(G\) a finite linear group. Let \(R=S^G\), then \(S\) is a CT \(R\)-module. The main result for CT modules is that for \(R\) a normal \(3-s\)CY ring, (1) CT modules are those reflexive generators which give NCCRs, (2)\(R\) has a NCCR\(\Longleftrightarrow R\) has a NCCR given by a CM generator \(\Longleftrightarrow R\) has a CT module.
After giving precise definitions of modifying and maximal modifying models the authors prove that when a NCCR exists, then the Maximal modifying algebras are exactly the same as NCCRs.
The authors explain their results on derived equivalences. They show that any algebra derived equivalent to a modifying algebra also has the form \(\text{End}_R(M)\). They give a result detailing the relationship between modifying and maximal modifying modules on the level on categories, and a relationship between maximal modifying modules and tilting modules. A main issue is the construction of modifying modules by mutations, by a method using exchange sequences. Defining left and right mutations allows the mutation of any NCCR of any dimension, at any direct summand, and gives another NCCR together with a derived equivalence, and this process together with its main results are covered in the article.
This is a long and advanced article. However, it is rather explicit with nice examples, and this makes it a good introduction and survey of AR-duality and its generalization. The article rather self contained, and illustrates the development of the categorical representation theory. noncommutative Crepant resolutions; Auslander Reiten theory; Auslander Reiten duality; Auslander Reiten algebra; cluster tilting; mutations; right mutation; left mutation; exchange seqences O. Iyama and M. Wemyss, Maximal modifications and Auslander-Reiten duality for non-isolated singularities, Invent. Math. 197 (2014), no. 3, 521-586. Noncommutative algebraic geometry, Representations of quivers and partially ordered sets, Local theory in algebraic geometry Maximal modifications and Auslander-Reiten duality for non-isolated singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper grew out of the author's work [J. Reine Angew. Math. 375/376, 47-66 (1987; Zbl 0628.14023)] on the interpretation of deformations of simple elliptic singularities by singular del Pezzo surfaces. Since the cases arising are classified by subsystems of root systems, it seemed natural to seek direct relations between the geometry of the cases arising and properties of the root systems involved. In pursuing this, it is necessary to have at hand a complete list of the subsystems in question, so a direct proof of the classification of these subsystems was sought.
The paper starts by introducing basic notions, including those of \(\mathbb{Z}\)- and \(\mathbb{Q}\)-closed subsystems, then develops an algorithm in terms of the Coxeter diagram for listing all subsystems of a given root system.
In \S2 we classify subsystems up to equivalence by the Weyl group. This is easy to do directly for the classical cases and \(G_ 2\), so most time is spent on \(F_ 4\) and the \(E_ n\). For \(\mathbb{Q}\)-closed subsystems this is done by classifying subsets of the vertex set of the diagram by ``moves''. Analyzing the effect of these moves throws up a notion ``strict on the right'' which reappears in \S3.
We then present various situations (simple and simple-elliptic singularities; intersections of two quadrics) where deformations of singularities can be described in terms of root systems, and we seek to relate the geometry to the combinatorics. In particular, where we have a curve \(\Gamma\), we can characterize the cases when \(\Gamma\) is reducible.
It is well known that the root systems \(B_ l\), \(C_ l\), and \(F_ 4\) can be obtained by ``folding'' \(A_{2l-1}\), \(D_{l+1}\), and \(E_ 6\) respectively. In \S4 we give an axiomatic description of this process which allows us to investigate its effects on subsystems. We then proceed to geometrical applications analogous to those above. algorithm for subsystems of a root system; simple elliptic singularities; singular del Pezzo surfaces; Coxeter diagram; intersections of two quadrics; deformations of singularities C.\ T.\ C. Wall, Root systems, subsystems and singularities, J. Alg. Geom. 1 (1992), 597-638. Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Computational aspects of algebraic surfaces, Deformations of complex singularities; vanishing cycles Root systems, subsystems and 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 Over fields of characteristic zero, resolution of singularities is achieved by means of an inductive argument, which is sustained on the existence of the so called hypersurfaces of maximal contact. We report here on an alternative approach which replaces hypersurfaces of maximal contact by generic projections. Projections can be defined in arbitrary characteristic, and this approach has led to new invariants when applied to the open problem of resolution of singularities over arbitrary fields. We show here how projections lead to a form of elimination of one variable using invariants that, to some extent, generalize the notion of discriminant.
This exposition draws special attention on this form of elimination, on its motivation, and its use as an alternative approach to inductive arguments in resolution of singularities. Using techniques of projections and elimination one can also recover some well known results. We illustrate this by showing that the Hilbert-Samuel stratum of a d-dimensional non-smooth variety can be described with equations involving at most d variables.
In addition this alternative approach, when applied over fields of characteristic zero, provides a conceptual simplification of the theorem of resolution of singularities as it trivializes the globalization of local invariants. resolution of singularities; Rees algebras Bravo, A; Villamayor U, OE, Elimination algebras and inductive arguments in resolution of singularities, Asian J. Math., 15, 321-355, (2011) Global theory and resolution of singularities (algebro-geometric aspects), Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics Elimination algebras and inductive arguments in resolution of singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this paper the author studies the configuration of simple singularities in a reduced sixtic curve in \(\mathbb{P}^2\). Recall that singularities are described by their dual graph in an embedded resolution of singularities and simple singularities are in correspondence with simple Dynkin diagrams, of type \(A_i\) \((i\geq 1)\), \(D_j\) \((j\geq 4)\) and \(E_k\) \((k=6,7,8)\). The main theorem in this article is:
There exists a reduced sixtic curve in \(\mathbb{P}^2\) with only simple singularities given by a finite Dynkin graph \(G\) if and only if \(G\) is a subgraph of a graph listed by the author.
For the proof the author uses full version of Nikulin's embedding theorem of even lattices. resolution of singularities; Dynkin diagrams; sixtic curve Jin-Gen Yang, Sextic curves with simple singularities, Tôhoku Math. J. 48 (1996), 203--227. Singularities of curves, local rings, Global theory and resolution of singularities (algebro-geometric aspects), Computational aspects of algebraic curves Sextic curves with simple singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(\varepsilon, \delta \geq 0\). A pair \((X, B)\) over an algebraically closed field of characteristic zero is called \((\varepsilon, \delta)\)-log canonical at \(x \in X\) if it is \(\varepsilon\)-lc near \(x\) and there exists a \(\delta\)-plt blowup of \((X, B)\) at \(x\). The first condition means that every prime divisor over \(X\) has log discrepancy \(\geq \varepsilon\). A \(\delta\)-plt blowup \(\pi \colon Y \to X\) extracts a unique \(\pi\)-anti-ample exceptional divisor \(E\) over \(x\), is an isomorphism otherwise and if \(K_Y + B_Y = \pi^*(K_X + B)\), then any exceptional divisor over \(Y\) has log discrepancy \(> \delta\) with respect to \(K_Y + B_Y + E\). Let \(D_n(\varepsilon, \delta)\) be the class of all \((\varepsilon, \delta)\)-lc singularities in dimension \(n\).
Theorem 1.1: Let \(\varepsilon, \delta > 0\). Then \(D_n(\varepsilon, \delta)\) is bounded up to deformation, i.e.~any singularity in this class can be deformed (over \(\mathbb A^1\)) to a singularity parametrized by a fixed scheme \(S\) of finite type.
Corollary 1.2: Over the complex numbers, \(D_2(\varepsilon, \delta)\) is analytically bounded.
Theorem (rather, Corollary) 1.3: Let \(i(x \in X)\) be an upper semicontinuous invariant of klt singularities. Then \(i\) is bounded from above on \(D_n(\varepsilon, \delta)\). Lower semicontinuous invariants are similarly bounded from below.
Corollaries 1.4 and 1.5: The multiplicity and (over \(\mathbb C\)) the embedding dimension are bounded on \(D_n(\varepsilon, \delta)\). klt singularities; deformations; log discrepancies; bounded families; plt blow-up Minimal model program (Mori theory, extremal rays), Singularities in algebraic geometry Bounded deformations of \((\epsilon,\delta)\)-log canonical singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(K\), \(K_\infty\), \(A\), \(C\) and \(\Gamma\) be respectively a global field \(K\) of positive characteristic, its completion \(K_\infty\) at a fixed place \(\infty\), the ring \(A\) of elements of \(K\) regular outside \(\infty\), the completion \(C\) of an algebraic closure of \(K_\infty\) and an arithmetic subgroup \(\Gamma\) of \(GL_2 (K)\), i.e. a subgroup commensurable with \(GL_2 (A)\). Set \(\Omega= C-K_\infty\). This is a Drinfeld upper half-plane, \(\Gamma\) acts by fractional linear transformations on \(\Omega\) and the rigid analytic space \(M_\Gamma= \Gamma\setminus \Omega\) is indeed an affine curve over \(C\); this is a Drinfeld modular curve. These curves, for various \(\Gamma\), are the substitutes in positive characteristic of the classical modular curves.
Let \(\overline{M}_\Gamma\) be the canonical completion of \(M_\Gamma\). The author studies divisors on \(\overline{M}_\Gamma\) with support in the set of cusps, i.e. in \(\overline{M}_\Gamma- M_\Gamma\) (cuspidal divisors). To use analytic methods [as in \textit{E.-U. Gekeler} and \textit{M. Reversat}, J. Reine Angew. Math. 476, 27-93 (1996; Zbl 0848.11029)], in \S 2, the author generalizes the theory of theta functions developed in (loc. cit.) to ``degenerate parameters''. One knows that the period lattice (in the sense of Manin-Drinfeld) of \(\overline{M}_\Gamma\) is described by a set of harmonic cochains or a set of theta functions (loc. cit.). The author adds to that interpretation the description of the links between new theta functions, cuspidal divisors and harmonic cochains (\S 3). In \S 4, where \(\Gamma\) is assumed to be a congruence subgroup, the author studies, with the help of the preceding methods, the canonical map between the group \({\mathcal C}(\Gamma)\) generated in the Jacobian \({\mathcal J}_\Gamma\) of \(\overline{M}_\Gamma\) by the cusps (the group \({\mathcal C}(\Gamma)\) is finite here), and the group \(\Phi_\infty (\Gamma)\) of connected components of the Néron model of \({\mathcal J}_\Gamma\) at \(\infty\). Finally (\S 5), in the case of \(A= \mathbb{F}_q [T]\) and for a Hecke congruence subgroup, the author obtains more information about his new theta functions and the (eventually non-empty) kernel of the map \({\mathcal C}(\Gamma)\to \Phi_\infty(\Gamma)\). class group; Drinfeld upper half-plane; Drinfeld modular curve; theta functions; cuspidal divisors; harmonic cochains; congruence subgroup E.-U. Gekeler,On the cuspidal divisor class group of a Drinfeld modular curve, Documenta Mathematica. Journal der Deutschen Mathematiker-Vereinigung2 (1997), 351--374. Drinfel'd modules; higher-dimensional motives, etc., Arithmetic aspects of modular and Shimura varieties, Holomorphic modular forms of integral weight, Modular and Shimura varieties On the cuspidal divisor class group of a Drinfeld modular 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 This article uses techniques from algebraic geometry and homological algebra, together with ideas from string theory to construct a class of 3-dimensional Calabi-Yau algebras which are non-commutative crepant resolutions of Gorenstein affine toric threefolds. A characteristic property of an \(n\)-dimensional Calabi-Yau manifold \(X\) is that the \(n\)th power of the shift functor on \(D(X):=D^b(\mathrm{coh}(X))\), the bounded derived category of coherent sheaves on \(X\), is a Serre functor. That is, there exists a natural isomorphism
\[
\text{Hom}_{D(X)}(A,B)\cong\text{Hom}_{D(X)}(B,A[n])^\ast, \;\forall A,B\in D(X).
\]
The idea behind Calabi-Yau algebras is to write down conditions on the algebra \(A\) such that \(D(A):=D^b(\mathrm{mod}(A))\), the bounded derived category of modules over \(A\), has the same property.
One way to study resolutions of singularities is by considering their derived categories. Of particular interest are crepant resolutions of toric Gorenstein singularities. It is conjectured by Bondal and Orlov that if \(f_1:Y_1\rightarrow X\) and \(f_2:Y_2\rightarrow X\) are crepant resolutions then there is a derived equivalence \(D(Y_1)\cong D(Y_2)\). This means that the derived category of a crepant resolution is an invariant of the singularity.
A tilting bundle \(T\) is a bundle which determines a derived equivalence \(D(Y)\cong D(A)\), where \(A=\text{End}(T)\). In this case one could consider the algebra \(A\) as a type of noncommutative crepant resolution (NCCR) of the singularity.
Given a NCCR \(A\), a (commutative) crepant resolution \(Y\) such that \(D(Y)\cong D(A)\) can be constructed as a moduli space of certain stable \(A\)-representations. This is a generalization of the McKay correspondence.
If \(X=\text{Spec}(R)\) is a Gorenstein singularity, then any crepant resolution is a Calabi-Yau variety. Therefore, if \(A\) is an NCCR it must be a Calabi-Yau algebra. The center of \(A\) must also be the coordinate ring \(R\) of the singularity. Thus any 3-dimensional Calabi-Yau algebra whose center \(R\) is the coordinate ring of a toric Gorenstein 3-fold, is potentially an NCCR of \(X=\text{Spec}(R)\).
Bocklandt proved that every graded Calabi-Yau algebra of global dimension 3 is isomorphic to a superpotential algebra. Such an algebra \(A=\mathbb C Q/(dW)\) is the quotient of the path algebra of a quiver \(Q\) by an ideal of relations \((dW)\), where the relations are generated by taking (formal) partial derivatives of a single element \(W\) called the superpotential. It encodes some informations about the syzygies. The Calabi-Yau condition is actually equivalent to saying that all the syzygies can be obtained from the superpotential.
To construct algebras from Calabi-Yau manifolds, the author uses the concept of dimer models. A dimer model is a finite bipartite tiling of a compact (oriented) Riemann surface \(Y\). Of particular interest are tilings of the 2-torus where the dimer model can be considered as a doubly periodic tiling of the plane. A dual tiling is also considered, where faces are dual to vertices and edges dual to edges. The edges of this dual tiling inherit an orientation from the bipartiteness of the dimer model. This is usually chosen so that the arrows go clockwise around a face dual to a white vertex. Therefore the dual tiling is a quiver \(Q\), with faces. The faces of this quiver encodes a superpotential \(W\), and so there is a superpotential algebra \(A=\mathbb C Q/(dW)\) associated to every dimer model. Of special importance are the perfect matchings in a dimer model used to construct a commutative ring \(R=\mathbb C[X]\) from a dimer model. Then \(R\) is the coordinate ring in three variables of an affine toric Gorenstein 3-fold \(X\). The author describes conditions on the dimer model so that its superpotential algebra is Calabi-Yau. Consistency conditions are a strong type of non-degeneracy condition, and necessary and sufficient conditions for a dimer model to be geometrically consistent are given in terms of the intersection properties of special paths called zig-zag flows on the universal cover \(\tilde Q\) of the quiver \(Q\). Geometric consistency amaounts to saying that zig-zag flows behave effectively like straight lines.
The author study some properties of zig-zag flows in a geometrically consistent dimer model. He construct, in a very explicit way, a collection of perfect matchings indexed by the 2-dimensional cones in the global zig-zag fan. It is proved that these are all the perfect matchings. Also, it is seen that each perfect matching of this form corresponds to a vertex of multiplicity one.
The concept of (noncommutative, affine, normal) toric algebras are introduced. It is hoped that they might play a similar role in noncommutative algebraic geometry to that played by toric varieties in algebraic geometry. The author shows that there is a toric algebra \(B\) naturally associated to every dimer model, and moreover, the center of this algebra is the ring \(R\) associated to the dimer model in the way described above. Therefore a given dimer model has two noncommutative algebras \(A\) and \(B\) and there is a natural algebra homomorphism \(\mathfrak h:A\rightarrow B.\) A dimer model is called algebraically consistent if this map is an isomorphism. Algebraic and geometric consistency are the two consistency conditions studied in the core of the article. One essential result is that a geometrically consistent dimer model is algebraically consistent. The proof of this is heavily dependent on the definition of perfect matchings. The main results of the article is the following theorems:
1) If a dimer model on a torus is algebraically consistent, then the algebra \(A\) obtained from it is a Calabi-Yau algebra of global dimension 3.
2) Given an algebraically consistent dimer model on a torus, the algebra \(A\) obtained from it is an NCCR of the (commutative) ring \(R\) associated to that dimer model.
3) Every Gorenstein affine toric threefold admits an NCCR, which can be obtained via a geometrically consistent dimer model.
The book is explicit; it gives all the definitions, contains clear proofs, and is an excellent text on the subject. dimer model; Calabi-Yau algebra; superpotential algebra; non-commutative crepant resolutions; geometrically consistence; algebraically consistence Broomhead, N.: Dimer models and Calabi-Yau algebras. Mem. Am. Math. Soc. \textbf{215}(1011), viii+86 (2012) Research exposition (monographs, survey articles) pertaining to algebraic geometry, Toric varieties, Newton polyhedra, Okounkov bodies, Noncommutative algebraic geometry, Calabi-Yau manifolds (algebro-geometric aspects), Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics Dimer models and Calabi-Yau algebras | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities \textit{P. B. Kronheimer} [J. Differ. Geom. 29, 665-683 (1989; Zbl 0671.53045)] has constructed quiver varieties from extended Dynkin diagrams of types \(\widetilde A_n,\widetilde D_n,\widetilde E_n\). These quiver varieties are important objects for the study of simple singularities. Let \(p\) be the quotient map of the Cartan subalgebra by the Weyl group. The semiuniversal deformations of simple singularities are constructed on the quotient space of Cartan subalgebra by the Weyl group of corresponding types. Then these quiver varieties are the pull-back of semiuniversal deformations of simple singularities by the quotient map \(p\). These quiver varieties are obtained as the symplectic quotients of symplectic vector spaces by a reductive group. So the coordinate rings are invariant subrings of polynomial rings with respect to the action of the group. In general it is difficult to find a minimal set of generators of an invariant ring and the relations between them.
In this paper the author shows that this is possible for the case of quiver varieties constructed by P. B. Kronheimer. Moreover surprisingly it can be shown that the obtained relation is unique and irreducible. In this research the invariant theory of quivers by \textit{L. Le Bruyn} and \textit{C. Procesi} [Trans. Am. Math. Soc. 317, 585-598 (1990; Zbl 0693.16018)] and of matrices of low degrees by \textit{K. Nakamoto} [J. Pure Appl. Algebra 166, 125-148 (2002; Zbl 1001.15022)] is used. Omoda, Y.: On the coordinate ring of quiver varieties associated to extended Dynkin diagrams. J. math. Kyoto univ. 40, No. 2, 315-336 (2000) Lie algebras of linear algebraic groups, Group actions on affine varieties, Singularities of surfaces or higher-dimensional varieties, Actions of groups on commutative rings; invariant theory On the coordinate rings of quiver varieties associated to extended Dynkin diagrams | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this paper, the authors prove the \(A_{n}\) Dynkin type case of Reineke's conjecture, which states that ``If \(Q\) is a quiver of Dynkin type, there exists a weight system \(\Theta\) on \(Q\) such that the stable representations are precisely the indecomposable ones.'' To obtain an elementary combinatorial proof, they reformulate the conjecture claiming that it is equivalent to establish the following assertation:
\textit{Let \(Q\) be a Dynkin. Then the abelian category \(\mathrm{Rep}_{k}(Q)\) is a maximal stable category}
In other words, the \((w,r)\)-stable subcategory of \(\mathrm{Rep}_{k}(Q)\) consists of the indecomposable ones.
The proof of the main result is based on a so-called intrinsic system \(\Theta\) defined on the set of vertices \(Q_{0}\) of a quiver \(Q\) of Dynkin type \(A_{n}\), in the combinatorial proof of the main result, the authors classify vertices into four classes and for any indecomposable representation \(X\) an inequality of type \(\frac{w_{\Theta}(X')}{|Q^{X'}_{0}|}<\frac{w_{\Theta}(X)}{|Q^{X}_{0}|}\) is established for all the four cases and any subrepresentation \(X'\) of \(X\).
The paper finishes by giving a formula of the intrinsic system \(\Theta\) based on semi-invariant theory. The proof of this result is self-contained, so the authors recall the definitions of an affine variety associated with quiver representations of a given dimension and the corresponding ring of semi-invariants and the Euler product between dimension vectors. quiver representation; Reineke conjecture; semi-invariant; weight system Representations of quivers and partially ordered sets, Geometric invariant theory Stability and indecomposability of the representations of quivers of \(A_n\)-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 X be a quotient surface singularity, and define Gdef (X, r) as the directed graph of maximal Cohen-Macaulay (MCM) modules with edges corresponding to deformation incidences. We conjecture that the number of connected components of Gdef (X, r) is equal to the order of the divisor class group of X, and when X is a rational double point (RDP), we observe that this follows from a result of A. Ishii. We view this as an enrichment of the McKay correspondence. For a general quotient singularity X, we prove the conjecture under an additional cancellation assumption. We discuss the deformation relation in some examples, and in particular we give all deformations of an indecomposable MCM module on a rational double point. quotient singularities; modules; deformation theory Local deformation theory, Artin approximation, etc., Algebraic moduli problems, moduli of vector bundles, Cohen-Macaulay modules, Singularities of surfaces or higher-dimensional varieties The deformation relation on the set of Cohen-Macaulay modules on a quotient surface singularity | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The manuscript under review requires a minimal amount of knowledge of algebraic or complex geometry. Indeed, one can replace ``schemes'' with ``varieties'' or ``complex manifolds'' throughout the paper. Furthermore, the author provides many examples and exercises to support the process of understanding the main ideas and concepts. The theory of Donaldson-Thomas (DT) invariants associates integers to moduli spaces of stable sheaves on a compact Calabi-Yau \(3\)-fold. K. Behrend realized that these same numbers, originally written as integrals over algebraic cycles or characteristic classes, can also be obtained by an integral over a constructible function (which is called a Behrend function), with respect to the measure given by the Euler characteristic. This new perspective extends DT theory to non-compact moduli spaces, as well as ``motivic nature'' of this new invariant.
This manuscript aims to give a thorough discussion on DT theory in the case of quivers with potential. This results in omitting the role of orientation data as well as derived algebraic geometry arising in DT theory. However, many important ideas and concepts are still evident in the case of quiver representations that this paper is an excellent starting point to learn about DT theory. The paper begins with a notion of classifying objects, which form an abelian category. The author then goes into the concept of Artin and moduli stacks, in particular, of quotient stacks (Section \(2\), Example \(2.20\)), where a thorough discussion is given in Section \(2\).
We then study quiver moduli spaces and stacks in Section \(3\). Furthermore, the path algebra of a quiver is categorical, noncommutative analogue of a polynomial algebra in ordinary commutative algebra, and although we are introduced to quotients of the path category of a quiver by some ideal of relations, results in Section \(6.1\) only depend on the linear category. That is, a potential \(W\) is an element of the vector space \(\mathbb{C}Q/[\mathbb{C}Q,\mathbb{C}Q]\), where \([\mathbb{C}Q,\mathbb{C}Q]\) is the \(\mathbb{C}\)-linear span of all commutators (a potential can also be thought of as the \(0\)-th Hochschild homology of the \(\mathbb{C}\)-linear category \(\mathbb{C}Q\), or \(\mathbb{C}\)-linear combination of equivalence classes of cycles in a quiver \(Q\) where two cycles are equivalent if one can be transformed into the other by a cyclic permutation).
A constructible function is a function \(a:X(\mathbb{C})\rightarrow \mathbb{Z}\) on the set of closed points of a scheme \(X/\mathbb{C}\) of finite-type with only finitely-many values on each connected component of \(X\), and such that the level sets of \(a\) are the closed points of locally closed subsets in each connected component of \(X\). Introduced in Section \(4\), they can be pulled back and multiplied, and using fiberwise integrals with respect to the Euler characteristic, they can be pushed forward. Moreover, every locally closed subscheme determines a constructible (characteristic) function.
Motivic theory (see Section \(4.1\) and \(4.2\)) is a generalization of constructible functions, in that one associates to every scheme \(X\) an abelian group \(R(X)\) of functions on \(X\) which can be pulled back, pushed forward, and multiplied. In addition, there is a ``characteristic function'' in \(R(X)\), where \(X\) is a locally closed subscheme such that the characteristic function of a disjoint union is the sum of the characteristic function of its summands.
Vanishing cycles for schemes and quotient stacks are introduced in Section \(5\), which form an additional structure on stacky motivic theories formalizing the properties of ordinary classical vanishing cycles. For example, the Behrend function is a good example of a vanishing cycle on the theory of constructible functions. In Section \(6\), DT functions and invariants are introduced, where many examples are provided in Section \(6.2\) to illustrate the theory. We end this section in discussing Ringel-Hall algebras (Section \(6.3\)), an integration map (Section \(6.4\)), and a wall-crossing identity (Section \(6.5\)), which are some of the fundamental tools used in DT theory. Donaldson-Thomas theory; moduli stacks; constructible functions; quivers with potential; vanishing cycles; quotient stacks; Ringel-Hall algebra; wall-crossing; Grothendieck group of varieties Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Stacks and moduli problems, Representations of quivers and partially ordered sets, Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects), Intersection homology and cohomology in algebraic topology An introduction to (motivic) Donaldson-Thomas theory | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be a simply connected semisimple algebraic group over an algebraically closed field of characteristic zero and \(V(\lambda)\) be a simple \(G\)-module of highest weight \(\lambda\). Then \(\text{End}V(\lambda)\) is a simple \((G\times G)\)-module and \(\mathbb{P}(\mathrm{End}V(\lambda))\) is a \((G\times G)\)-variety. The orbit closure \(X_{\lambda}=\overline{(G\times G)[{\roman{id}}_{V(\lambda)}]}\subset\mathbb{P}(\text{End}V(\lambda))\) is an equivariant compactification of the image of \(G\) in \(\mathrm{PGL}(V(\lambda))\) considered as a symmetric space. The nature of singularities of the varieties \(X_{\lambda}\) is studied in the paper under review.
The following theorems are the main results of the paper: (A) \(X_{\lambda}\) is normal if and only if, whenever the support \(\text{Supp}(\lambda)\) of the highest weight \(\lambda\) contains a long simple root in a non-simply laced component of the Dynkin diagram of \(G\), it contains the short simple root which is adjacent to a long one in the component; (B) \(X_{\lambda}\) is smooth if and only if it is normal and the following three conditions hold:
(1) the intersection of \(\text{Supp}(\lambda)\) with each component of the Dynkin diagram is connected and is an extreme vertex of the component whenever it is a one-element set;
(2) \(\text{Supp}(\lambda)\) contains every junction vertex together with at least two neighboring vertices;
(3) the complement of \(\text{Supp}(\lambda)\) is a subdiagram having all components of type A.
These results specify general criteria for normality and smoothness of projective compactifications of reductive groups obtained by \textit{D. A. Timashev} [Sb. Math. 194, No. 4, 589--616 (2003); translation from Mat. Sb. 194, No. 4, 119--146 (2003; Zbl 1074.14043)]. A criterion for projective normality of \(X_{\lambda}\) was obtained by \textit{S. S. Kannan} [Math. Z. 239, No. 4, 673--682 (2002; Zbl 0997.14012)]: \(X_{\lambda}\) is projectively normal if and only if \(\lambda\) is minuscule. It should be noted that the criteria obtained in the paper are of pure combinatorial nature and easy to check. In fact, Theorems A and B are extended in the paper to a wider class of simple projective group compactifications \(X_{\Sigma}=\overline{(G\times G)[{\roman{id}}_V]}\subset\mathbb{P}(\text{End}V)\), where \(V=\bigoplus_{\lambda\in\Sigma}V(\lambda)\) is any (multiplicity-free) \(G\)-module and the set of highest weights \(\Sigma\) contains a unique maximal element \(\lambda_{\max}\) with respect to the dominance order. (``Simple'' means here ``having a unique closed orbit''.) In particular, it follows that \(X_{\Sigma}\) is smooth if and only if \(X_{\lambda_{\max}}\) is smooth.
Methods of the proofs include using wonderful compactifications of semisimple groups, results of Kannan on projective normality, Timashev's criterion for smoothness, etc. Further generalizations to completions of arbitrary symmetric spaces \(G/H\) in \(\mathbb{P}(V(\lambda))\) are also possible, but require different methods. semisimple algebraic groups; projective representations; group compactifications; wonderful varieties; symmetric spaces Bravi, Paolo; Gandini, Jacopo; Maffei, Andrea; Ruzzi, Alessandro, Normality and non-normality of group compactifications in simple projective spaces, Ann. Inst. Fourier (Grenoble), 0373-0956, 61, 6, 2435\textendash 2461 (2012) pp., (2011) Compactifications; symmetric and spherical varieties, Group actions on varieties or schemes (quotients), Homogeneous spaces and generalizations Normality and non-normality of group compactifications in simple projective 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 Resolution of singularities of surfaces over the field of complex numbers was first achieved by the \textit{J. E. Jung} in 1908. Jung's proof was based on the fact that the local fundamental group above a normal crossing of the branch locus is abelian. In 1955 [\textit{S. Abbyyankar}, Am. J. Math. 77, 575-592 (1955; Zbl 0064.27501)] it was shown that, in characteristic \(p>0\), such a local fundamental group may not even be solvable. In 1975 [\textit{S. Abhyankar}, ibid. 79, 825-856 (1957; Zbl 0087.03603)], this led to the conjecture that the algebraic fundamental group of the affine line over an algebraically closed ground field of characteristic \(p>0\) coincides with \(Q(p)\), where \(Q(p)\) is the set of all quasi-\(p\) groups, i.e., finite groups generated by their \(p\)-Sylow subgroups. In the 1957 paper it was also conjectured that the algebraic fundamental group of the (once) punctured affine line over an algebraically closed ground field of characteristic \(p>0\) coincides with \(Q_1(p)\), where \(Q_1(p)\) is the set of all cyclic-by-quasi-\(p\) groups, i.e., finite groups \(G\) such that \(G/p(G)\) is cyclic where \(p(G)\) is the subgroup of \(G\) generated by all of its \(p\)-Sylow subgroups. In support of these conjectures, in the 1957 paper, two families of equations were written down giving unramified coverings of the affine line and the punctured affine line, and it was suggested that their Galois groups be computed. These families were originally obtained by taking a section of the unsolvable surface covering of the 1955 paper.
In section 3 and 6, it now turns out that the Galois groups of these families include the alternating and symmetric groups \(A_n\) and \(S_n\) of any degree \(n>1\), the linear groups \(\text{GL}(m,q)\), \(\text{SL}(m,q)\), \(\text{PGL}(m,q)\), \(\text{PSL}(m,q)\), \(\text{AGL}(1,q)\) and \(\text{GF}(q)^+\) for any prime power \(q>1\) and any integer \(m>1\), the Mathieu groups \(M_{11}\), \(M_{12}\), \(M_{22}\), \(M_{23}\), and \(M_{24}\), and the group \(\Aut(M_{12})\).
In particular, the trinomial equation \(Y^{23}-XY^t+1=0\) has Galois group \(M_{23}\) for \(p=2\) and \(t=3\). In section 4, it is shown that for all other relevant values of \(p\) and \(t\) its Galois group is \(A_{23}\) or \(S_{23}\).
In section 5, a merging technique is explained which shows that, in the trinomial case, sometimes the two families can be converted into each other. In section 6, this together with reciprocation, leads to a list of equations having Mathieu groups as Galois groups. In section 7, there are proved some transitivity lemmas. In section 8, these are used to show that some other members of the first family also give unramified coverings of the affine line and the punctured affine line with Galois groups \(\text{GL}(m,q)\), \(\text{SL}(m,q)\), \(\text{PGL}(m,q)\) and \(\text{PSL}(m,q)\), and as a consequence there are deduced several explicit equations which have these groups as Galois groups and which give unramified coverings of the \(m\)-dimensional affine space or the \(m\)-dimensional affine space minus a hyperplane. characteristic \(p\); algebraic fundamental group of the affine line; Galois groups; Mathieu groups Shreeram S. Abhyankar, Mathieu group coverings and linear group coverings, Recent developments in the inverse Galois problem (Seattle, WA, 1993) Contemp. Math., vol. 186, Amer. Math. Soc., Providence, RI, 1995, pp. 293 -- 319. Coverings of curves, fundamental group, Separable extensions, Galois theory, Simple groups: alternating groups and groups of Lie type, Simple groups: sporadic groups Mathieu group coverings and linear group coverings | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 deformation theory of a two-dimensional singularity, which is isomorphic to the affine cone over a curve, is intimately linked with the (extrinsic) geometry of this curve. In recent times various authors have studied one-parameter deformations, partly under the guise of extensions of curves to surfaces. In this paper we consider the versal deformation of cones, in the simplest case: cones over hyperelliptic curves of high degree. In particular, we show that for degree \(4g + 4\), the highest degree for which interesting deformations exist, the number of smoothing components is \(2^{2g + 1}\) (except for \(g = 3)\). Powerful methods exist to compute \(T^1\) for surface singularities, without using explicit equations. In the homogeneous case the graded part \(T^1 (-1)\) is the hardest. We review in a general setting the relation with Wahl's Gaussian map [cf. \textit{J. Wahl}, J. Differ. Geom. 32, No. 1, 77-98 (1990; Zbl 0724.14022)]. We prove that \(T^1 (-1)\) vanishes for a cone over a general curve, embedded with an arbitrary linear system of degree at least \(2g + 11\). We find the dimension of \(T^2\) for hyperelliptic cones with the main lemma of a paper by \textit{K. Behnke} and \textit{J. A. Christophersen} [Compos. Math. 77, No. 3, 233-268 (1991; Zbl 0728.14034)], which connects the number of generators of \(T^2\) with the codimension of smoothing components in the versal base of a general hyperplane section. Therefore we compute \(T^1\) for the cone over \(d\) points on a rational normal curve of degree \(d - g - 1\). We use explicit equations for the curve singularity. Actually, the equations for the cone over a hyperelliptic curve have a nice structure. We give an interpretation of \(T^2 (-2)\) in terms of this structure.
Smoothing components are related to surfaces with \(C\) as hyperplane section. According to Castelnuovo, these are rational ruled. We get all from a given one by elementary transformations. An explicit description of the corresponding infinitesimal deformations enables us to conclude that the base space is a complete intersection of degree \(2^{2g + 1}\). We also consider smoothing data in the sense of \textit{E. Looijenga} and \textit{J. Wahl} [Topology 25, 261-291 (1986; Zbl 0615.32014)]. affine cone over a curve; versal deformation of cones; Wahl's Gaussian map; smoothing components; infinitesimal deformations Stevens, J.: Deformations of cones over hyperelliptic curves. J. reine angew. Math. 473, 87-120 (1996) Formal methods and deformations in algebraic geometry, Curves in algebraic geometry, Infinitesimal methods in algebraic geometry, Surfaces and higher-dimensional varieties Deformations of cones over hyperelliptic curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The present paper is devoted to prove the following statement:
Let \(G\cong 6.\mathrm{Suz}\) be the universal perfect central extension of the Suzuki simple group and \(U\) be a \(12\)-dimensional irreducible representation of \(G\). Then the quotient singularity \(U/G\) is weakly exceptional but not exceptional, in the sense of [\textit{V. V. Shokurov}, J. Math. Sci., New York 102, No. 2, 3876--3932 (2000; Zbl 1177.14078)] and [\textit{Yu. G. Prokhorov}, Blow-ups of canonical singularities. A. G. Kurosh, Moscow, Russia, May 25-30, 1998. Berlin: Walter de Gruyter. 301--317 (2000; Zbl 1003.14005)].
As consequence of this theorem, the authors get the following classification result: let \(G\) be a sporadic group or a central extension of one with centre contained in the commutator subgroup and let \(G\hookrightarrow \mathrm{GL}(U)\) be a faithful finite-dimensional complex representation of \(G\). Then the singularity \(U/G\) is: \begin{itemize}\item[-] exceptional if and only if \(G\cong 2.\mathrm{J}_2\) is a central extension of the Hall-Janko sporadic simple group and \(U\) is a \(6\)-dimensional irreducible representation of \(G\); \item[-] weakly exceptional but not exceptional if and only if \(G\cong 6.\mathrm{Suz}\) and \(U\) is a \(12\)-dimensional irreducible representation of \(G\).
This result shows that among the sporadic simple groups, the groups \(\mathrm{J}_2\) and \(\mathrm{Suz}\) are somehow distinguished from a geometric point of view. This motivates the author to pose the following question: ``Is there a group-theoretic property that distinguishes the groups \(\mathrm{J}_2\) and \(\mathrm{Suz}\) among the sporadic simple groups?''\end{itemize} weakly exceptional singularities; log canonical threshold; sporadic simple groups Singularities in algebraic geometry, Finite automorphism groups of algebraic, geometric, or combinatorial structures, Singularities of surfaces or higher-dimensional varieties, Minimal model program (Mori theory, extremal rays) Sporadic simple groups and quotient singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(( \mathfrak{g},\mathsf{g})\) be a pair of complex finite-dimensional simple Lie algebras whose Dynkin diagrams are related by (un)folding, with \(\mathsf{g}\) being of simply-laced type. We construct a collection of ring isomorphisms between the quantum Grothendieck rings of monoidal categories \(\mathscr{C}_{\mathfrak{g}}\) and \(\mathscr{C}_{\mathsf{g}}\) of finite-dimensional representations over the quantum loop algebras of \(\mathfrak{g}\) and \(\mathsf{g} \), respectively. As a consequence, we solve long-standing problems: the positivity of the analogs of Kazhdan-Lusztig polynomials and the positivity of the structure constants of the quantum Grothendieck rings for any non-simply-laced \(\mathfrak{g} \). In addition, comparing our isomorphisms with the categorical relations arising from the generalized quantum affine Schur-Weyl dualities, we prove the analog of Kazhdan-Lusztig conjecture (formulated in [\textit{D. Hernandez}, Adv. Math. 187, No. 1, 1--52 (2004; Zbl 1098.17009)]) for simple modules in remarkable monoidal subcategories of \(\mathscr{C}_{\mathfrak{g}}\) for any non-simply-laced \(\mathfrak{g} \), and for any simple finite-dimensional modules in \(\mathscr{C}_{\mathfrak{g}}\) for \(\mathfrak{g}\) of type \(\text{B}_n \). In the course of the proof we obtain and combine several new ingredients. In particular, we establish a quantum analog of \(T\)-systems, and also we generalize the isomorphisms of [\textit{D. Hernandez} and \textit{B. Leclerc}, J. Reine Angew. Math. 701, 77--126 (2015; Zbl 1315.17011); \textit{D. Hernandez} and \textit{H. Oya}, Adv. Math. 347, 192--272 (2019; Zbl 1448.17019)] to all \(\mathfrak{g}\) in a unified way, that is, isomorphisms between subalgebras of the quantum group of \(\mathsf{g}\) and subalgebras of the quantum Grothendieck ring of \(\mathscr{C}_{\mathfrak{g}} \). Finite-dimensional groups and algebras motivated by physics and their representations, Relationship to Lie algebras and finite simple groups, Simple, semisimple, reductive (super)algebras, Grothendieck groups, \(K\)-theory, etc., Affine algebraic groups, hyperalgebra constructions, Quantum groups and related algebraic methods applied to problems in quantum theory, Quantum groups (quantized enveloping algebras) and related deformations, Ring-theoretic aspects of quantum groups Isomorphisms among quantum Grothendieck rings and propagation of positivity | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 goal of the paper is to classify all finite subgroups \(G\) of \(\mathrm{Sp}(V)\) for which \(V/G\) admits a symplectic resolution. Let \(V\) be a symplectic vector space and \(G \subset \mathrm{Sp}(V)\) a finite group. The authors study singularities of the quotient \(V/G\). If there exists a projective resolution of singularities \(X \to V/G\) such that \(X\) is a symplectic manifold, then they say that \(V/G\) admits a projective symplectic resolution. Consider \(V\) as a symplectically irreducible representation of \(G\). The authors classify all such pairs \((V,G)\), which admit a projective symplectic resolution, in the case \(\dim V \neq 4\), except for four singularities, occurring in dimensions at most 10, for which the question remains open.
Section 2 contains the definition of symplectic variety and symplectic resolutions, and some criteria are given for the (non)existence of projective symplectic resolutions. In Section 3, they recall the Kleinian group, in Section 4, Cohen's classification of symplectic reflection groups. Section 5 deals with two general criteria for the non-existence of projective symplectic reflections for the group \(G(K,H,\alpha)\) with \(H \neq {1}\). In the next section, they prove that the symplectically imprimitive, irreducible symplectic reflection groups of the above type satisfy at least one of the criteria. Section 7 gives the following result: If \(n>2\), then the symplectic quotient \(C^{2n}/G_n(K,H)\) admits a projective symplectic resolution if and only if \(K=H\). The paper ends with some open questions. symplectic resolution; symplectic smoothing; symplectic reflection algebra; Poisson variety; quotient singularity; McKay correspondence Bellamy, G.; Schedler, T., On the (non)existence of symplectic resolutions of linear quotients, Math. Res. Lett., 23, 1537-1564, (2016) Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, McKay correspondence, Deformations of associative rings, Poisson algebras On the (non)existence of symplectic resolutions of linear 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 Gabrielov numbers describe certain Coxeter-Dynkin diagrams of the 14 exceptional unimodal singularities and play a role in Arnold's strange duality. In a previous paper, the authors defined Gabrielov numbers of a cusp singularity with an action of a finite abelian subgroup \(G\) of \(\mathrm{SL}(3,\mathbb C)\) using the Gabrielov numbers of the cusp singularity and data of the group \(G\). Here we consider a crepant resolution \(Y \to \mathbb C^3/G\) and the preimage \(Z\) of the image of the Milnor fibre of the cusp singularity under the natural projection \(\mathbb C^3 \to\mathbb C^3/G\). Using the McKay correspondence, we compute the homology of the pair \((Y,Z)\). We construct a basis of the relative homology group \(H_3(Y,Z;\mathbb Q)\) with a Coxeter-Dynkin diagram where one can read off the Gabrielov numbers. cusp singularity; group action; crepant resolution; McKay correspondence; Coxeter-Dynkin diagram; Gabrielov numbers Complex surface and hypersurface singularities, Milnor fibration; relations with knot theory, McKay correspondence, Group actions on varieties or schemes (quotients) A geometric definition of Gabrielov numbers | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In his previous work [\textit{W. Wang}, Duke Math. J. 103, 1--23 (2000; Zbl 0947.19004)], the author has indicated the deep analogy and connections between (A) the theory of the Hilbert scheme \(X^{[n]}\) of \(n\) points on a (quasi-) projective surface \(X\) [cf. e.g. \textit{H. Nakajima}, Lectures on Hilbert schemes of points on surfaces (University Lecture Series 18, Providence, RI: AMS)(1999; Zbl 0949.14001)], and (B) the theory of wreath products \(\Gamma_{[n]} =\Gamma^n\rtimes S_n\) between a power \(\Gamma^n\) of a finite group \(\Gamma\) and the symmetric group \(S_n\) [cf. e.g. \textit{A. Zelevinsky}, Representations of finite classical groups. A Hopf algebra approach (Lecture Notes in Mathematics 869, Berlin: Springer) (1981; Zbl 0465.20009)]. The reviewed article is a survey that is related to many questions and works of this topic: first of all, the construction of the Heisenberg and Virasoro algebras in the framework of (A), with description of the cohomology ring of \(X^{[x]}\) and in the framework of (B), with description of vertex representations of affine and toroidal Lie algebras, whose Dynkin diagrams correspond to finite subgroups \(\Gamma\) of \(\text{SL}_2(\mathbb{C})\), a.o.
If \(Y\) is a quasi-projective surface acted by a finite group \(\Gamma\), and a resolution of singularities \(X\to Y/\Gamma\) (1) is given, then, according to the author [loc.cit.], one obtains the resolution of singularities \(X^{[n]} \to Y^n/\Gamma_{[n]}\) (2) that can preserve many properties of the resolution (1), in particular, some of them that are concerned with the Euler and Hodge numbers of the corresponding orbifolds.
Under some additional conditions, equivalence between the bounded derived categories \(D_{\Gamma_{[n]}}(Y^n)\) and \(D(X^{[n]})\) of \(\Gamma_{[n]}\)-equivariant coherent sheaves on \(Y^n\) and coherent sheaves on \(X^{[n]}\) respectively is established. The case \(Y= \mathbb{C}^2\) is considered in more detail, in particular, the question of replacement of \(X^{[n]}\) (in case of a certain known minimal \(X\) in (1)) by a special subvariety \(Y_{\Gamma,n}\) of \((\mathbb{C}^2)^{[nN]}\) (where \(N\) is the order of \(\Gamma\)) that admits a quiver description.
Finally, a short dictionary for comparison of analogous notions in the theories (A) and (B) is presented. algebraic surfaces; action of finite groups; resolution of singularities; Heisenberg algebra; Virasoro algebra; representation of Lie algebras; vertex algebras; equivariant \(K\)-theory of schemes W. Wang, Algebraic structures behind Hilbert schemes and wreath products, in: S. Berman et al. (Eds.), Recent Developments in Infinite-Dimensional Lie Algebras and Conformal Field Theory, Charlottesville, VA, 2000; Contemp. Math. 297 (2002) 271-295. Parametrization (Chow and Hilbert schemes), Virasoro and related algebras, Vertex operators; vertex operator algebras and related structures, Extensions, wreath products, and other compositions of groups Algebraic structures behind Hilbert schemes and wreath products. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 and simply connected algebraic group. The authors compute the Picard group of the moduli stack of quasi-parabolic \(G\)-bundles over a smooth, complete complex curve \(X\). Quasi parabolic \(G\)-bundles are defined with respect to \(n\) distinct points of \(X\), each one labelled by a parabolic subgroup of \(G\) containing the same Borel subgroup. The proof requires the uniformization theorem which describes the stack as double quotient of certain infinite dimensional algebraic groups. Basic facts about stacks and Lie theory needed in the proofs are clearly presented in this paper. Generators for the Picard group are explicitly computed when \(G\) is classical or \(G_2\), by constructing a pfaffian line bundle. This construction is tricky and requires explicit computations.
An application is the construction of a square root of the dualizing bundle of the stack for \(n=0\). The authors find also a canonical isomorphism between the space of global sections of the above stack and the corresponding space of conformal blocks of Tsuchiya, Ueno and Yamada which appears in conformal field theory. A consequence of this isomorphism is a generalization of the Verlinde formula. For an account about the Verlinde formula and its relation to mathematical physics see the survey of \textit{C. Sorger} [Sém. Bourbaki, Vol. 1994/95, Astérisque 237, 87-114, Exp. No. 794 (1996; Zbl 0878.17024)].
At the end the authors determine that the Picard group of the moduli space of \(G\)-bundles over curves is isomorphic to \({\mathbb{Z}}\), a result found independently by \textit{S. Kumar} and \textit{M. S. Narasimhan} [Math. Ann. 308, No. 1, 155-173 (1997; Zbl 0884.14004)]. For \(G=SL(r)\) this is a classical result of Drezet and Narasimhan. The assumption of simple connectedness of \(G\) has been removed in a subsequent paper [see \textit{A. Beauville, Y. Laszlo} and \textit{C. Sorger}, Composit. Math. 112, No. 2, 183-216 (1998)]. moduli stack; parabolic bundles over a complex curve; Picard group; uniformization; pfaffian line bundle; conformal blocks; Verlinde formula Y.~Laszlo, C.~Sorger, The line bundles on the moduli of parabolic \(G\)-bundles over curves and their sections, {\em Ann.~Sci.~Éc.~Norm.~Supér.~(4)} 30(4): 499--525, 1997 Picard groups, Algebraic moduli problems, moduli of vector bundles, Vector bundles on curves and their moduli The line bundles on the moduli of parabolic \(G\)-bundles over curves and their sections | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 goal of this work is to present evidence that singularities of the variety associated to the ring of invariants may be the critical factor determining local triviality of fixed point free \(G_{a}\) action.'' Here \(G_{a}\) is the additive group of complex numbers.
The authors study derivations \(D\) of \(\mathbb{C}[x,y,z,\omega ]\) that are twin triangular, that is if \(D(\omega)=0\), \(D(z)\in \mathbb{C}[\omega ]\), \(D(y)\in \mathbb{C} [z,\omega ],\) and \(D(x)\in \mathbb{C}[z,\omega ].\) The examples cited in the paper of actions on \(\mathbb{C}^{4}\) and \(\mathbb{C}^{5}\) [taken from \textit{J. K. Deveney, D. R. Finston}, and \textit{M. Gehrke}, Commun. Algebra 22, No. 12, 4977-4988 (1994; Zbl 0817.14029); \textit{J. K. Deveney} and \textit{D. R. Finston}, Proc. Am. Math. Soc. 123, No. 3, 651-655 (1995; Zbl 0832.14036); \textit{J. Winkelmann}, Math. Ann. 286, No. 1, 593-612 (1990; Zbl 0708.32004)] are all generated by simple twin triangular derivations.
Let \(D\) be a twin triangular derivation with finitely generated kernel, such that the associated \(G_{a}\) action has no fixed points and the ring of \(G_{a}\) invariants is finitely generated. Assume furthermore that \(D(z)\) has no multiple roots. Under these conditions, \(D\) is shown to be GICO, that is they satisfy the concept of geometric irreducibility in codimension one.
Two classes of twin triangular actions are introduced. One class consists of the derivations with \(D(z)=\omega \) and the other consists of derivations where \(D(x),D(y)\in \mathbb{C} [z].\) It is shown that the action \(G_{a}\times \mathbb{C}\rightarrow \mathbb{C}\) that corresponds to a derivation of either class is either conjugate to a translation with quotient isomorphic to \(\mathbb{C}^{3}\) or is not proper (hence not locally trivial). derivations; twin triangular derivations; actions on \(\mathbb{C}^n\); ring of invariants; twin triangular actions J.K. Deveney and D.R. Finston: Twin triangular derivations , Osaka J. Math. 37 (2000), 15--21. Group actions on varieties or schemes (quotients), Actions of groups on commutative rings; invariant theory, Automorphism groups of \(\mathbb{C}^n\) and affine manifolds Twin triangular derivations | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \(\sigma =\sigma_ K=\sigma_{{\mathbb{Q}}(\sqrt{- 3})}={\mathbb{Z}}+{\mathbb{Z}}\omega\), \(\omega =e^{\pi i/3}\), be the ring of integral Eisenstein numbers and \(\Gamma =U((2,1),\sigma)\) be a lattice in U((2,1),\({\mathbb{C}})\). Denote by \(\Gamma '\) the congruence subgroup of \(\Gamma\) with respect to the prime ideal \((1-\omega)\sigma\). Then \(\Gamma/\Gamma' \cong \sigma_ 4\) the symmetric group of 4 letters.
Let B be the complex 2-ball, let \((B/\Gamma)\) be the Baily-Borel-Satake compactification of the open algebraic surface \(B/\Gamma\) and let \(K(B/\Gamma)\) be the field of K-Picard modular functions. A singular module on B is an isolated fixed point of (elements of) the group \(GU((2,1),K)=\{\gamma \in GL_ 3({\mathbb{C}})|\) \(<\gamma a,\gamma b>=c_{\gamma}<a,b>\) for all \(a,b\in {\mathbb{C}}^ 3\}\) where \(<.,.>\) denotes the Hermitian scalar product on \({\mathbb{C}}^ 3\) with signature (2,1).
The main result of the paper is that for any singular module \(\sigma\) and any K-Picard modular function \({\mathcal F}\), \({\mathcal F}(\sigma)\) is an algebraic number. arithmetic points; algebraicity; integral Eisenstein numbers; Picard modular function Holzapfel, R.-P. : An arithmetic uniformization for arithmetic points of the plane by singular moduli , J. Ramanujan Math. Soc. 3(1), (1988), S.35-62. Theta series; Weil representation; theta correspondences, Global ground fields in algebraic geometry, Families, moduli, classification: algebraic theory An arithmetic uniformization for arithmetic points of the plane by singular moduli | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(P \in X\) be a Kawamata log terminal singularity. \(P \in X\) is called exceptional if and only if for any effective \(\mathbb{Q}\)-divisor \(D\) such that the pair \((X,D)\) has log canonical singularities, there exists at most one exceptional divisor \(E\) with discrepancy \(a(E,X,D)=-1\). \(P \in X\) is called weakly exceptional if and only if there exists a unique birational map \(f : Y \rightarrow X\) such that the exceptional set of \(f\) is an irreducible divisor \(E\), \((Y,E)\) has purely log terminal singularities, \(-E\) is \(f\)-ample and \(P \in f(E)\). Minimal model theory shows that every exceptional singularity is weakly exceptional.
This paper classifies finite subgroups \(G < \mathrm{GL}(n,\mathbb{C})\), \(n \leq 4\), such that the quotient \(X=\mathbb{C}^n/G\) has weakly exceptional singularities. The case \(n=2\) was treated by V. Shokurov who showed that the two dimensional exceptional singularities are exactly the DuVal singularities of type \(D_n, E_6, E_7, E_8\).
The classification method used by the author is the following. Let \(\bar{G}\) be the image of \(G\) in \(\mathrm{PGL}(n-1,\mathbb{C})\). Then according to a result of I. Cheltsov and C. Shramov, \(\mathbb{C}^n/G\) has weakly exceptional singularities if and only if \(\mathrm{lct}(\mathbb{P}^{n-1},\bar{G})\geq 1\), where \(\mathrm{lct}(\mathbb{P}^{n-1},\bar{G})\) is the global \(\bar{G}\)-invariant log canonical threshold of \(\mathbb{P}^{n-1}\) [\textit{I. Cheltsov} and \textit{C. Shramov}, Geom. Topol. 15, No. 4, 1843--1882 (2011; Zbl 1232.14001)]. Then by lifting \(\bar{G}\) to \(\mathrm{SL}(n,\mathbb{C})\), one can assume that \(G\) is a subgroup of \(\mathrm{SL}(n,\mathbb{C})\). Then according to a result of I. Cheltsov and C. Shramov, \(\mathrm{lct}(\mathbb{P}^{n-1},\bar{G})\geq 1\) if and only if \(G\) does not have semi invariants of degree at most \(n-1\) for its action on \(\mathbb{C}^n\), \(n \leq 4\). Hence if \(\mathbb{C}^n/G\) does not have weakly exceptional singularities, \(G\) fixes a degree at most \(n-1\), \(n \leq 4\), hypersurface of \(\mathbb{P}^{n-1}\) and therefore it is a subgroup of the automorphisms of either a conic, a quadric surface or a cubic surface. Then the classification follows from the explicit classification of finite subgroups of \(\mathrm{SL}(n,\mathbb{C})\) and the automorphism groups of cubic surfaces, quadrics and conics. exceptional singularities; weakly exceptional singularities; log canonical; log terminal; plt; quotient singularities D Sakovics, Weakly-exceptional quotient singularities Singularities in algebraic geometry, Birational automorphisms, Cremona group and generalizations, General theory for finite permutation groups, Primitive groups, Finite automorphism groups of algebraic, geometric, or combinatorial structures Weakly-exceptional 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 review is extracted from the introduction of the article.
``In their foundational paper [\textit{L. A. Borisov} et al., J. Am. Math. Soc. 18, No. 1, 193--215 (2005; Zbl 1178.14057)], Borisov, Chen, and Smith introduce the notion of a \textit{stacky fan}, the combinatorial data from which one constructs toric Deligne-Mumford (DM) stacks, which are the stack-theoretic analogues of classical toric varieties. When the corresponding fan is polytopal, classical toric varieties have been studied from the perspectives of both algebraic and symplectic geometry. Similarly, when the underlying fan of a stacky fan is polytopal, a toric DM stack admits a description in the language of symplectic geometry via the combinatorial data of a \textit{stacky polytope} introduced by Sakai [\textit{H. Sakai}, Result. Math. 63, No. 3--4, 903--922 (2013; Zbl 1286.57026)]. (In the symplectic-geometric context -- and particularly in this manuscript -- stacks are considered over the category of smooth manifolds.)
The mathematical contributions of this manuscript are as follows. We first describe in Theorem 2.2, Proposition 2.4, and Proposition 2.15 an explicit computation of the isotropy groups of toric DM stacks, realized as quotient stacks \([Z/G]\) for appropriate space \(Z\) and abelian Lie group \(G\), in terms of the combinatorial data (i.e. stacky fan) determining the toric DM stack.
Secondly, as an application of our description of isotropy groups of toric DM stacks, we give a computation of the connected component of the identity element \(G_0 \subset G\) and the component group \(G/G_0\) in terms of the underlying stacky fan (Proposition 2.11, Lemma 2.8, Proposition 2.17). In this manuscript, our computation of isotropy groups leads to a characterization of those toric DM stacks that are (stacks equivalent to) global quotients of a finite group action and to a description of its universal cover (cf. Section 2.2).
Our third set of results concern weighted projective stacks (resp. `\textit{fake weighted projective stacks}'), which are natural stack-theoretic analogues of the classical weighted projective spaces (resp. fake weighted projective spaces). These form a rich class of examples that have been studied extensively both as stacks and as orbifolds. As another application of our computation of isotropy groups, in Proposition 3.2 (resp. Proposition 3.4) we obtain an exact characterization of those stacky polytopes which yield (stacks equivalent to) weighted projective stacks (resp. fake weighted projective stacks).
Finally, in Section 4 we introduce a class of labelled polytopes, which we call \textit{labelled sheared simplices}. These are labelled simplices with all facets but one lying on coordinate hyperplanes. In this special case we illustrate the aforementioned results concretely in terms of the facet labels.'' stacky fan; toric Deligne-Mumford stack; inertia group; weighted projective spaces R. Goldin, M. Harada, D. Johannsen and D. Krepski: Inertia groups of a toric DM stack, fake weighted projective spaces, and labelled sheared simplices , to appear in Rocky Mountain J. Math., arXiv: Toric varieties, Newton polyhedra, Okounkov bodies, Topology and geometry of orbifolds, Geometric invariant theory Inertia groups of a toric Deligne-Mumford stack, fake weighted projective stacks, and labeled sheared simplices | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 simple complex Lie group \({{G}^{\mathbb{C}}}\), the authors study integrable systems of the Hitchin type constructed for moduli spaces of quasi-compact Higgs \({{G}^{\mathbb{C}}}\)-bundles over singular curves. The notion of quasi-compact structure is introduced by the authors to mean that the gauge group at the marked points on the base spectral curve is reduced to a maximal compact subgroup \(K\). Note that in the original case studied by \textit{N. Hitchin} [Duke Math. J. 54, 91--114 (1987; Zbl 0627.14024)] describing the moduli space of \({{G}^{\mathbb{C}}}\)-Higgs bundles as the phase space of complex integrable systems, the gauge group at the marked points was reduced to parabolic subgroups. However, in this quasi-compact setting the part of coordinates in the local description of the Higgs bundles is real and the authors implement a specific real involution on the moduli space in order to construct a \textit{real completely integrable system} as the fixed point set of this involution.
Interesting examples of such integrable systems are further discussed, called Models I, II and III in the article, as generalizations of the spin Calogero-Sutherland model related to \({{G}^{\mathbb{C}}}\) with two types of interacting spin variables. Model I is a free system defined on the symplectic space \({{T}^{*}}{{G}^{\mathbb{C}}}\times {{T}^{*}}K\); Model II on \({{T}^{*}}{{G}^{\mathbb{C}}}\) and Model III on \({{T}^{*}}{{G}^{\mathbb{C}}}\times {{T}^{*}}{{{\mathcal{X}}}^{\mathbb{C}}}\), where \({{{\mathcal{X}}}^{\mathbb{C}}}={{{G}^{\mathbb{C}}}}/{K}\;\) the Riemannian symmetric space. In all these cases, the Lax operators of the systems are explicitly constructed using the quasi-compact structure of the Higgs bundles. quasi-compact Higgs bundle; Calogero-Sutherland system; real Hamiltonian system; Lax operator Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills), Global differential geometry of Hermitian and Kählerian manifolds, Kähler-Einstein manifolds, Simple, semisimple, reductive (super)algebras, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Vector bundles on curves and their moduli Quasi-compact Higgs bundles and Calogero-Sutherland systems with two types of spins | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 second edition (see [Zbl 1308.14001] for a review of the first edition) is extended by a chapter on recent developments.
After a short motivating first chapter that introduces the questions addressed in this book, follow two chapters dealing with background, one on sheaves, algebraic varieties and analytic spaces, and one on homological algebra and duality. The treatment includes spectral sequences. In these chapters the theorems are formulated in their natural generality. The definition of a singularity is initially given both for analytic spaces and for algebraic varieties over an arbitrary algebraically closed field. After stating Artin's Algebraization Theorem, that an isolated singularity of an analytic space is isomorphic to the germ of an algebraic variety over \(\mathbb{C}\), only the algebraic case is considered. After a chapter defining the canonical divisor for varieties over an arbitrary algebraically closed field the further discussion is restricted to the field of complex numbers.
The book defines log canonical, canonical, log terminal, terminal and rational singularities and provides a characterization of isolated such ones in terms of plurigenera. The classification is refined in the two-dimensional case, and rational surface singularities are described in some detail. Also the results of the Author on two-dimensional Du Bois singularities are introduced. The next chapter considers the analogous
questions for higher dimensional singularities, and in particular for the case of dimension three. It concludes with the list of the famous
95 families of simple \(K3\)-singularities.
The final chapter presents some developments after the publication of the first Japanese version [Zbl 1308.14002] of this book. These concern log discrepancies for pairs and the use of the space of arcs in their description. This opens up the possibility of proving results in positive characteristic. The book closes with a brief introduction to \(F\)-singularities, in positive characteristic. rational singularities; minimal model program; Du Bois singularities; arc spaces; F-singularities Research exposition (monographs, survey articles) pertaining to algebraic geometry, Singularities in algebraic geometry, Deformations of singularities, Global theory and resolution of singularities (algebro-geometric aspects), Local complex singularities Introduction to 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 quiver \(Q\) is a directed graph specified by a set of vertices \(Q_0\), a set of arrows \(Q_1\), and head and tail maps which situate each arrow. Many moduli spaces in algebraic geometry and gauge theory parameterize objects that may be described as quiver bundles, i.e. as representations of quivers in the category of coherent sheaves. The construction and analysis of the moduli spaces invariably involves a careful study of extensions and deformations of the objects being parameterized. This in turn hinges on the hypercohomology of a certain deformation complex. Special cases include Higgs bundles [see \textit{I. Biswas} and \textit{S. Ramanan}, J. Lond. Math. Soc. 49, 219--231 (1994; Zbl 0819.58007)], vortices or holomorphic pairs [\textit{M. Thaddeus}, Invent. Math. 117, 317--353 (1994; Zbl 0882.14003)] and holomorphic triples [\textit{S. Bradlow, O. Garcia-Prada} and \textit{P. Gothen}, J. Differ. Geom. 64, 111--170 (2004; Zbl 1070.53054)].
The purpose of the paper under review is to isolate the abstract structures common to all the examples, and to establish the essential results in a general setting. To this end the authors study quiver representations in abelian categories. In particular, they consider the case of representations in sheaves of \(\mathcal O_X\) modules on an algebraic variety \(X\). They allow the sheaves at the tail of each arrow in the quiver to be twisted by a fixed sheaf; they call the resulting objects twisted \(Q\)-sheaves.
Using an explicit injective resolution of a twisted \(Q\)-sheaf they show that there is long exact sequence which relates the Ext groups for the twisted \(Q\)-sheaf to the usual sheaf Ext groups of the constituent sheaves in the twisted \(Q\)-sheaf. They show further that if \(V\) and \(W\) are twisted \(Q\)-sheaves with \(V\) locally free, then the groups \(\text{Ext}^p(V,W)\) are isomorphic to hypercohomology groups \(\mathbb{H}^p(C(V,W))\), where \(C(V,W)\) is a natural complex of sheaves associated to the twisted \(Q\)-sheaves. In specific examples, this complex corresponds to the deformation complex which controls the local structure of the moduli spaces; the results of this paper thus specialize to the standard deformation theory results mentioned above. Gothen P.B., King A.D., Homological algebra of twisted quiver bundles, J. London Math. Soc., 2005, 71(1), 85--99 Algebraic moduli problems, moduli of vector bundles, Ext and Tor, generalizations, Künneth formula (category-theoretic aspects), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Representations of quivers and partially ordered sets Homological algebra of twisted quiver bundles | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be a finite group acting on a quasi-projective smooth scheme \(X\) over a field \(k\). Suppose that \(g:\tilde{Y}\to X/G\) is a resolution of singularities of \(X/G\). Then by the McKay principal many aspects of the geometry of \(\tilde{Y}\) is reflected in the \(G\)-equivariant geometry of \(X\). Assume that \(E=g^{-1}(Z)_{\text{red}}\) where \(Z \subset X/G\) is the singular locus is a divisor with simple normal crossing. Denote by \(\Gamma(E)\) the dual complex associated to \(E\). For example, in the case of the Klein singularity \(\mathbb{C}^2/G\), \(\Gamma(E)\) is one of the ADE Dynkin diagrams.
The paper under review studies the homotopy type of \(\Gamma(E)\) which is known to be independent of the choice of the resolution. To express the result let \(\pi:X\to X/G\) be the projection, \(T=g^{-1}(Z)_{\text{red}}\), and \(f:\tilde{X}\to X\) be a proper birational \(G\)-equivariant morphism such that \(\tilde{X}\) is a smooth \(G\)-scheme and \(E_T=f^{-1}(T)_{\text{red}}\) is a \(G\)-strict simple normal crossing divisor and \(f\) is an isomorphism over \(X-T\). The main result of the paper under review proves that if \(k\) is perfect with characteristic zero, then there is a canonical map \(\phi: \Gamma(E_T)/G\to \Gamma(E)\) in the homotopy category of \(CW\)-complexes which induces isomorphisms on the homology and fundamental groups. This result has important implications such as
1) \(\Gamma(E_T)/G\) is contractible \(\Rightarrow\) \(\Gamma(E)\) is contractible.
2) \(X/G\) have isolated singularities \(\Rightarrow\) T is smooth \(\Rightarrow\) \(\Gamma(E)\) is contractible.
The question of the contractibility of \(\Gamma(E)\) is very important, for example if \(X/G\) has rational singularities then \(\Gamma(E)\) is contractible. The proof of the main result regarding the fundamental group uses a geometric interpretation of the fundamental group by means of the classifying group of \(cs\)-coverings of \(E\). And regarding the homology groups the proof is based on an equivariant weight homology theory introduced in the paper under review, and proving an analog of McKay principal for this weight homology. The proof of the latter relies on introducing another (arithmetic) homology theory in the paper under review called equivariant Kato homology. The other ingredients of the proof are cohomological Hasse principal, Deligne's theorem on Weil conjecture and Gabber's refinement of De Jong's alteration theorem. McKay principal; Equivariant geometry Global theory and resolution of singularities (algebro-geometric aspects), McKay correspondence, Geometric class field theory Cohomological Hasse principle and resolution of quotient singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Considering complex \(n\)-dimension Calabi-Yau homogeneous hypersurfaces \({\mathcal H}_n\) with discrete torsion and using the Berenstein and Leigh algebraic geometry method, we study fractional D-branes that result from stringy resolution of singularities. We first develop the method introduced by \textit{D. Berenstein} and \textit{R. G. Leigh} [J. High Energy Phys. 2001, No. 6, Paper 030 (2001), see also Preprint hep-th/0105229] and then build the non-commutative (NC) geometries for orbifolds \({\mathcal O}={\mathcal H}_n/ \mathbb{Z}^n_{n+2}\) with a discrete torsion matrix \(t_{ab}=\exp[\frac {i2\pi}{n+2} (\eta_{ab}-\eta_{ba})]\), \(\eta_{ab}\in SL(n,\mathbb{Z})\). We show that the NC manifolds \({\mathcal O}^{(nc)}\) are given by the algebra of functions on the real \((2n+4)\) fuzzy torus \({\mathcal T}_{\beta_{ij}}^{2(n+2)}\) with deformation parameters \(\beta_{ij}= \exp \frac{i2 \pi}{n+2} [(\eta_{ab}^{-1}- \eta^{-1}_{ba}) q^a_iq_j^b]\) with \(q_i^a\) being charges of \(\mathbb{Z}^n_{n+2}\). We also give graphic rules to represent \({\mathcal O}^{(nc)}\) by quiver diagrams which become completely reducible at orbifold singularities. It is also shown that regular points in these NC geometries are represented by polygons with \((n+2)\) vertices linked by \((n+2)\) edges while singular ones are given by \((n+2)\) non-connected loops. We study the various singular spaces of quintic orbifolds and analyse the varieties of fractional D-branes at singularities as well as the spectrum of massless fields. Explicit solutions for the NC quintic \({\mathcal Q}^{(nc)}\) are derived with details and general results for complex \(n\)-dimensional orbifolds with discrete torsion are presented. fractional \(D\)-branes; Calabi-Yau homogeneous hypersurfaces; stringy resolution; discrete torsion Saidi, E. H.: NC geometry and fractional branes. Class. quantum grav. 20, 4447-4472 (2003) Noncommutative geometry methods in quantum field theory, String and superstring theories in gravitational theory, Calabi-Yau manifolds (algebro-geometric aspects), Relationships between surfaces, higher-dimensional varieties, and physics, String and superstring theories; other extended objects (e.g., branes) in quantum field theory NC geometry and fractional branes. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 thesis under review may be viewed as a step forward in understanding the semi-universal deformation of complex analytic isolated complete intersection singularities. One main question is: given such a singularity, which singularities does it deform to ? The answer depends on a detailed knowledge of the geometry of the deformation, and the author does an exhaustive study for simple singularities on space curves. These were classified by \textit{M. Giusti} [Singularities, Summer. Inst., Arcata/Calif. 1981, Proc. Symp. Pure Math. 40, Part I, 457-494 (1983; Zbl 0525.32006)].
The main result is that for the \(S_{\mu}, T_ 7, T_ 8\) and \(T_ 9\) singularities (Giusti's notation), the base space of the semi-universal deformation is isomorphic to a quotient X/W of a torus embedding X by a Weyl group W, such that the discriminant of the group action maps to the discriminant of the deformation. This parallels other known descriptions of the discriminant, for example as done by \textit{E. Brieskorn}, in Actes Congr. intern. Math. (1970), part 2, 279-284 (1981; Zbl 0223.22012) for simple hypersurface singularities [see also \textit{P. Slodowy}, ''Simple singularities and simple algebraic groups,'' Lect. Notes Math. 815 (1980; Zbl 0441.14002)].
The novelty is the introduction of torus embeddings which are constructed from extended Dynkin diagrams corresponding to generalized root systems. This construction, based on recent work by \textit{E. Looijenga} [Invent. Math. 61, 1-32 (1980; Zbl 0436.17005)], is done from a general point of view and should be applicable to other situations. The explicit study of the deformations uses classical algebraic geometry, e.g. results on families of hyperelliptic curves and del Pezzo surfaces. semi-universal deformation of complex analytic isolated complete; intersection singularities; simple singularities on space curves; torus embeddings; Dynkin diagrams; root systems; semi-universal deformation of complex analytic isolated complete intersection singularities Deformations of singularities, Singularities of curves, local rings, Deformations of complex singularities; vanishing cycles, Formal methods and deformations in algebraic geometry, Complete intersections Torus embeddings and deformations of simple singularities and space 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 is a translation of the book ``Le problème des modules pour les branches planes'' of \textit{O. Zariski} [Hermann, Paris (1973; Zbl 0317.14004)] by Ben Lichtin. It is based on notes from a course of Zariski at Centre de Mathématiques de l'École Polytechnique 1973 and contains an appendix by B. Teissier considering the moduli problem from the point of view of deformation theory. Zariski's aim is to study the space of isormorphism classes of plane curve singularities (analytically irreducible curve germs) of given equsingularity type, i.e. the moduli space \(M(\Gamma)\) of plane curve singularities with fixed semigroup \(\Gamma\). His ideas and results were the basis for further research in this direction [cf. for example \textit{O. A. Laudal} and the reviewer, ``Local moduli and singularities'', Lect. Notes Math. 1310 (1988; Zbl 0657.14005)].
The first three chapters of the book introduce the basic notions, especially invariants as the semigroup, the conductor, the characteristic of a branch, short parametrizations, etc. Then the moduli space \(M(\Gamma)\) is studied. It is proved that \(M(\Gamma)\) is not seperated in general. The structure of \(M(\Gamma)\) is analyzed for special examples. The dimension of the generic component of the moduli space for \(\Gamma=\langle n, n+1\rangle\) is computed. This was generalized later on by C. Delorme.
The appendix of Teissier is based on the idea that each plane curve singularity with semigroup \(\Gamma\) appears as a deformation of the associated monomial curve, more precise in its negatively weighted part. It turns out that every plane branch has a miniversal equisingular deformation, over a smooth base, with an equisingular section. Among several results a ``natural'' compactification of \(M[\Gamma)\) is given, as well as an interpretation of the generic component. moduli problem; plane curve singularity; semi group; deformation Zariski, O., \textit{The Moduli Problem for Plane Branches (with an Appendix by Bernard Teissier)}, 39, (2006), AMS, Providence RI Singularities of curves, local rings, Research exposition (monographs, survey articles) pertaining to algebraic geometry The moduli problem for plane branches. With an appendix by Bernard Teissier. Transl. from the French by Ben Lichtin | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 good resolution of an isolated singularity \(0\) of an algebraic (or analytic) variety \(X\) is one where the exceptional locus is a divisor \(Z\) with simple normal crossings. Such a good resolution is available in characteristic zero. Moreover, one can find one where all the intersections involving components of \(Z\) are irreducible (property I). To a good resolution of an isolated singularity \(x \in X\) one may associate a CW-complex \(\Gamma (Z)\) of dimension \(\leq \dim X - 1\), the dual complex associated to the resolution. Its homotopy type is independent of the chosen good resolution. If this satisfies condition I then \(\Gamma (Z)\) is a simplicial complex. Working over the complex numbers, in this article the author proves the following results about this object.
Theorem 1. If \(0 \in X\) is an isolated rational singularity (i.e., for a resolution \(f:Y \to X\), \({R^i}_{\star} {\mathcal O}_{Y}=0, ~ i>0\)), \(\dim X = n\), then \(H^{n-1}({\Gamma}(Z), C)=0\).
Theorem 2. If \(0\) is an isolated singularity of a hypersurface the \({\Gamma}(Z)\) has the homotopy type of a point.
As a corollary of these results it is proved that \(\Gamma (Z) \) is homotopically equivalent to a point under any one of the following assumptions: either \(0 \in X\) is an isolated rational singularity of dimension 3. or \(0 \in X\) is a 3-dimensional Gorenstein terminal singularity. The proof of Theorem 1 generalizes an argument of \textit{M. Artin}'s seminal paper on rational singularities of surfaces [Am. J. Math. 88, 129--136 (1966; Zbl 0142.18602)], but also involves a technical lemma on the degeneracy of a spectral sequence associated to a divisor with normal crossings in a Kähler manifold. The second theorem cleverly uses the fact that the link of an isolated hypersurface singularity of dimension at least three is simply connected. The paper is well written and has good bibliographical references. rational singularity; hypersurface singularity; good resolution of singularities; dual complex; cohomology; fundamental group Stepanov, D.A.: A note on resolution of rational and hypersurface singularities. Proc. Am. Math. Soc. \textbf{136}(8), 2647-2654 (2008) Singularities in algebraic geometry, Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants, Local theory in algebraic geometry, Modifications; resolution of singularities (complex-analytic aspects) A note on resolution of rational and hypersurface singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The article develops results concerning the projective McKay correspondence for the action of a finite group \(\widetilde{G}\subset \text{PSL}(2,{\mathbb C})\) on \({\mathbb P}^1,\) analogous to those obtained by \textit{A. Kirrilov} [Mosc. Math. J. 6, No. 3, 505--529 (2006; Zbl 1171.14302)]. More precisely, let \(G\subset\text{SL}(2,{\mathbb C})\) be a group such that \(\widetilde{G}/{\pm I}=G,\) where \(I\) is the identity matrix. Let \(\Gamma\) be the affine Dynkin graph associated to \(G\) and let \(\Pi_\Gamma\) denote the preprojective algebra. There is an equivalence between the bounded derived category of \(\widetilde{G}-\)equivariant coherent sheaves on the cotangent bundle \(T^*{\mathbb P}^1\) and the bounded derived category of finitely generated \(\Pi_\Gamma-\)modules. This provides a link with the \(2-\)dimensional affine McKay correspondence (see \textit{T. Bridgeland, A. King, M. Reid} [J. Am. Math. Soc. 14, No.3, 535--554 (2001; Zbl 0966.14028)]). That is, there exists a sequence of equivalences
\[
D^b_{\widetilde{G}}(T^*{\mathbb P}^1)\simeq D^b(\Pi_\Gamma)\simeq D^b_G({\mathbb C}^2)\simeq D^b(Y),
\]
where \(Y\) is the minimal resolution of \({\mathbb C}^2/G.\)
The equivalence \(R\Phi_h:D^b_{\widetilde{G}}(T^*{\mathbb P}^1)\longrightarrow D^b(\Pi_\Gamma)\) constructed in the article, depends on a height function \(h.\) Such function assigns an integral number to each vertex of graph \(\Gamma\) and every two height functions are connected by a series of Bernstein-Gelfand-Ponomarev reflections, [cf. \textit{I. N. Bernstein, I. M. Gelfand} and \textit{V. A. Ponomarev}, Russ. Math. Surv. 28, No.2, 17-32 (1973; Zbl 0279.08001)].
Let \({\mathcal D}\) denote the subcategory of \(D^b_{\widetilde{G}}(T^*{\mathbb P}^1)\) consisting of objects supported along the zero section. Define \({\mathcal B}_h\) to be the heart associated to the t-structure obtained by restricting the pull-back of the standard t-structure on \(D^b(\Pi_\Gamma)\) to the category \({\mathcal D}.\) Theorem 5.8 shows that hearts \({\mathcal B}_h, {\mathcal B}_{\sigma_i^{\pm} h},\) where \({\sigma_i^{\pm} h}\) denotes the height function obtained by reflection of \(h\) at the vertex \(i\) of \(\Gamma\), are related by a spherical twist. Moreover, the spherical twist between the hearts can be realized as tilting (Theorem 5.8 and Proposition 5.9).
The proof of above results involves the theory of Koszul algebras. In particular, it is shown that the preprojective algebra \(\Pi_\Gamma\) is the Koszul dual of the \(\text{Ext}-\)algebra of a \(\Gamma-\)collection of spherical objects in \(D^b_{\widetilde{G}}(T^*{\mathbb P}^1)\).
Section 3 of the article contains a detailed review of the results by A. Kirillov. McKay correspondence; Dynkin quiver; quiver representations; cotangent bundle of projective line; Koszul duality; reflection functor; spherical twist; derived equivalence McKay correspondence, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Representations of quivers and partially ordered sets, Quadratic and Koszul algebras The projective McKay correspondence | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We revisit the construction of elliptic class given by Borisov and Libgober for singular algebraic varieties. Assuming torus action we adjust the theory to the equivariant local situation. We study theta function identities having a geometric origin. In the case of quotient singularities \(\mathbb{C}^n/G\), where \(G\) is a finite group the theta identities arise from McKay correspondence. The symplectic singularities are of special interest. The Du Val surface singularity \(A_n\) leads to a remarkable formula. theta function; McKay correspondence; elliptic class of singular varieties; quotient singularities Local complex singularities, Singularities in algebraic geometry Elliptic classes, McKay correspondence and theta identities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 zero and \(\mathbb{X}= \text{GL}_N/P\) a flag variety over \(k\). It is a homogeneous variety under the action of \(\text{GL}_N\). We attach to the parabolic subgroup \(P\) an infinite, locally finite and levelled quiver. Then there exist relations \({\mathcal R}\) in the quiver \(Q\), such that the following four categories are equivalent: the category of \(\text{GL}_N\)-bundles on \(\mathbb{X}\), the category of finite dimensional representations of the group \(P\), the category of finite dimensional representations of the Lie-algebra \({\mathfrak p}\) of \(P\), and the category of finite dimensional representations of the quiver \(Q\), which satisfy the relations \({\mathcal R}\). If \(P\) is a maximal parabolic subgroup, then \(\mathbb{X}\) is a Grassmannian and the relations are quadratic. It is well known, that exceptional vector bundles admit a \(\text{GL}_N\)-structure. Because the category of \(\text{GL}_N\)-bundles has a richer structure than the category of all vector bundles, we are interested in this category to attack the problem of classification of all exceptional vector bundles on \(\mathbb{X}\).
In the first section of this article we compute the simple objects in the category of \(\text{GL}_N\)-bundles (proposition 1.1.18), and compute the terms of a minimal projective resolution of the simple \(\text{GL}_N\)-modules (theorem 1.2.7). The four equivalent categories do not have projective objects. So we embed them in some larger category and prove, that the projective representations in the larger category are exactly the restricted inverse limits of shifted \(\text{GL}_N\)-modules (theorem 1.2.9). In the second section we consider three examples, the three-dimensional flag variety \(\mathbb{F} (1,2;k^3)\), the two-dimensional projective space \(\mathbb{P}^2\), and the product of two projective lines \(\mathbb{P}^1 \times \mathbb{P}^1\). We note, that the classification of exceptional bundles is known on \(\mathbb{P}^2\) and \(\mathbb{P}^1 \times \mathbb{P}^1\) by results of J. M. Drezet and J. Le Potier and A. N. Rudakov. On \(\mathbb{F}(1,2;k^3)\) we construct two exceptional bundles of the same rank, but with different dimension vector under shift, duality and reflection. Further we compute the Cartan matrix and the dimension vectors of the simple \(\text{GL}_N\)-modules. classification of exceptional vector bundles; flag variety; quiver : Examples of Distinguished Tilting Sequences on Homogeneous Varieties, Canadian Math. Soc., Conference Proceedings, Vol. 18 (1996) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Representations of quivers and partially ordered sets, Grassmannians, Schubert varieties, flag manifolds, Representation theory of groups Examples of distinguished tilting sequences 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, the author continues the study of Gorenstein three dimensional singularities coming from finite group quotients. The article of \textit{S. S.-T. Yau} and \textit{Y. Yu} [``Gorenstein quotient singularities in dimension three'', Mem. Am. Math. Soc. 505 (1993; Zbl 0799.14001)] describes the action of \(G\subset SL(3, \mathbb{C})\), a finite group, on \(\mathbb{C}^3\) and classifies the Gorenstein quotients which arise out of this procedure, by classifying the finite groups of the necessary type. The author's work is towards settling the following conjecture:
Let \(G\) be a finite subgroup of \(SL(3, \mathbb{C})\) acting on \(\mathbb{C}^3\). Then there exists a resolution of singularities \(\sigma: \widetilde X\to \mathbb{C}^3/G\) with \(\omega_{\widetilde X} = {\mathcal O}_{\widetilde X}\) and \(\chi (\widetilde X)= \#\)\{conjugacy classes of \(G\}\).
This is apparently of some interest to the physicists. Various cases of this conjecture had been proved earlier and the author gives a summary of what was known. The author had settled the case of trihedral groups [\textit{Y. Ito}, Proc. Japan Acad., Ser. A 70, No. 5, 131-136 (1994; Zbl 0831.14006) and Int. J. Math. 6, No. 1, 33-43 (1995; Zbl 0831.14005)]. In the present article more cases are settled from types (B) and (D), as opposed to the trihedral groups which are of type (C), in the classification table of Yau and Yu. The author also mentions a later preprint of \textit{S.-S. Roan} [Inst. Math., Acad. Sinica, preprint R940606-1 (1994)] where the rest of the cases which were not covered earlier are treated, thereby proving the conjecture in full. action of group on complex 3-space; Gorenstein three dimensional singularities; finite group quotients; resolution of singularities Ito, Y.: Gorenstein quotient singularities of monomial type in dimension three. J. math. Sci. univ. Tokyo 2, 419-440 (1995) Singularities in algebraic geometry, Group actions on varieties or schemes (quotients), Modifications; resolution of singularities (complex-analytic aspects), Low codimension problems in algebraic geometry, Homogeneous spaces and generalizations, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), \(3\)-folds, Global theory and resolution of singularities (algebro-geometric aspects) Gorenstein quotient singularities of monomial type in dimension three | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We present a local computation of deformations of the tangent bundle for a resolved orbifold singularity \(\mathbb{C}^d/G\). These correspond to \((0,2)\)-deformations of \((2,2)\)-theories. A McKay-like correspondence is found predicting the dimension of the space of first-order deformations from simple calculations involving the group. This is confirmed in two dimensions using the Kronheimer-Nakajima quiver construction. In higher dimensions such a computation is subject to nontrivial worldsheet instanton corrections and some examples are given where this happens. However, we conjecture that the special crepant resolution given by the \(G\)-Hilbert scheme is never subject to such corrections, and show this is true in an infinite number of cases. Amusingly, for three-dimensional examples where \(G\) is abelian, the moduli space is associated to a quiver given by the toric fan of the blow-up. It is shown that an orbifold of the form \(\mathbb{C}^3 / \mathbb{Z}_7\) has a nontrivial superpotential and thus an obstructed moduli space. Aspinwall, P. S.: A mckay-like correspondence for (0,2)-deformations String and superstring theories; other extended objects (e.g., branes) in quantum field theory, McKay correspondence, Toric varieties, Newton polyhedra, Okounkov bodies, Global theory and resolution of singularities (algebro-geometric aspects), Parametrization (Chow and Hilbert schemes), Deformations of singularities, Topology and geometry of orbifolds, Representations of quivers and partially ordered sets, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Blow-up in context of PDEs A McKay-like correspondence for \((0,2)\)-deformations | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The two papers under review deal with the same problem: the classification of three-dimensional exceptional log canonical hypersurface singularities, which is a problem of birational geometry that arises in the log minimal model program. The first paper classifies the ``well-formed'' singularities and the second deals with the others. The first paper [Izv. Math. 66, 949--1034 (2002; Zbl 1076.14049)] is the main, more important one.
This classification is a positive answer, in the three-dimensional case, to the conjecture that there are only a finite number of types of \(d\)-dimensional exceptional log canonical hypersurface singularities, for any \(d \geq 3\).
The first paper also contains the solution to the above problem for \(d = 2\). First, let us describe this 2-dimensional classification, which is more familiar: a 2-dimensional log terminal singularity is exceptional if and only if it belongs to one of the types \(\mathbb E_6, \mathbb E_7, \mathbb E_8\); a 2-dimensional strictly log canonical singularity is exceptional if and only if it is either simply elliptic or it belongs to one of the types \(\widetilde{\mathbb D}_4, \widetilde{\mathbb E}_6, \widetilde{\mathbb E}_7, \widetilde{\mathbb E}_8\). (A description of 3-dimensional exceptional strictly log canonical hypersurface singularities is also given in the first paper).
The essential difference between the 2-dimensional and the \(d\)-dimensional case, \(d\geq 3\), is that almost every type of multidimensional singularity contains an infinite number of non-isomorphic singularities with the same resolution.
Reviewer's remark. The classification of the singularities, divided into 12 tables, is very detailed and very long (with more than 900 singularities); the list takes up about 51 of the 85 pages of the first paper. The author proves that there is a finite number of types of these singularities as a consequence of the finite nature of the classification list, but no mention is made of the classification into different types (as in the above 2-dimensional case). Be that as it may, it is easy to understand the effort that went into this involved classification: for instance, the author needed 17 definitions to classify the singularities in the first paper. log minimal model program; \(d\)-dimensional exceptional log canonical hypersurface singularities \(3\)-folds, Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Hypersurfaces and algebraic geometry Classification of three-dimensional exceptional log-canonical hypersurface singularities. II. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper under review studies the monodromy conjecture for symplectic resolutions as formulated by Braverman-Maulik-Okounkov. This conjecture roughly says that the monodromy of the quantum connection for two birationally equivalent smooth symplectic Deligne-Mumford stacks which are symplectic resolutions of a singular symplectic stack and they correspond to the two large radius points in the compactified Kähler moduli spaces is the same as the monodromy given by their equivalence in the \(K\)-theory.
This paper introduces the notion of ``extended stacky hyperplane arrangements'' and defines the hypertoric DM stacks associated with them. There is a one-to-one correspondence between such arrangements and the GIT data and in particular the wall crossing of hypertoric DM stacks gives the Mukai type flops.
Let \(X_+ \to X_-\) be a crepant birational map between two smooth Lawrence toric DM stacks given by a single wall crossing. It is known that \(X_+\) and \(X_-\) are derived equivalent and their equivalence is given by the Fourier-Mukai transformation, which in the level of \(K\)-theory matches the analytic continuation of the \(I\)-function, and so also the analytic continuation of the quantum connections which are determined by the \(I\)-function. The wall crossing above implies that the associated birational transformation \(Y_+\to Y_-\) for the hypertoric DM stacks is crepant, and the Fourier-Mukai functor gives an equivalence of derived categories of \(Y_+\) and \(Y_-\). The paper under review proves that this Fourier-Mukai transformation matches the analytic continuation of quantum connections of the hypertoric DM stacks, which is induced by the analytic continuation
of the associated Lawrence toric DM stacks.
Viewing crepant birational transformation of hypertoric DM stacks as the local model of stratified Mukai type flops for general symplectic DM stacks, one can say that the construction in this paper will play a role in the study for general Mukai type flops by degeneration method to the local models. crepant transformation conjecture; monodromy conjecture Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Generalizations (algebraic spaces, stacks), McKay correspondence The crepant transformation conjecture implies the monodromy 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 Simple singularities (also called ADE because the different types \(A_ k\), \(D_ k\) and \(E_ 6\), \(E_ 7\), \(E_ 8\) in which they are classified) appear in a natural way in problems of classification of singularities from different points of view. In dimension 1, simple singularities have been characterized in terms of their resolution procedure by \textit{W. P. Barth}, \textit{C. A. M. Peters} and \textit{A. J. H. M. Van de Ven} [``Compact complex surfaces'', Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Band 4 (1984; Zbl 0718.14023)] in characteristic 0 and in any characteristic by \textit{K. Kiyek} and \textit{G. Steinke} [Arch. Math. 45, 565-573 (1985; Zbl 0553.14012)] who obtained the normal forms of them.
In dimension 2 simple singularities are just the rational double points [see \textit{M. Artin}'s paper in Complex Anal. Algebr. Geom., Collect. Papers dedic. K. Kodaira, 11-22 (1977; Zbl 0358.14008)]. On the other hand in characteristic 0 and arbitrary dimension \textit{V. I. Arnol'd} showed [cf. Funct. Anal. 6(1972), 254-272 (1973); translation from Funkts. Anal. Prilozh. 6, No.4, 3-25 (1972; Zbl 0278.57011)] that they are exactly the hypersurface singularities of finite deformation type (i.e. singularities from which one can obtain, by deformation, only a finite number of nonequivalent singularities).
According to previous works of \textit{H. Knörrer} [Invent. Math. 88, 153- 164 (1987; Zbl 0617.14033)] and \textit{R.-O. Buchweitz}, \textit{G.-M. Greuel} and \textit{F.-O. Schreyer} [ibid. 165-182 (1987; Zbl 0617.14034)] it is known that the simple singularities are the hypersurface singularities with finite Cohen-Macaulay type (i.e. singularities for which there exists only a finite number of non isomorphic indecomposable maximal Cohen-Macaulay modules over their local ring). However, the relationship with the point of view of Arnol'd (using deformation theory) remains essentially unknown in arbitrary characteristic.
This point constitutes the main result in the paper, more precisely, if \(f\in k[[x_ 1,...,x_ n]]\), k being an algebraically closed field of arbitrary characteristic, the authors define f to be simple if and only if f is contact-equivalent with one of the normal forms given in dimension 1 by Kiyek and Steinke, in dimension 2 by Artin and for dimension greater than two by double suspension of curves or surfaces.
Then, the main theorem asserts that the following statements are equivalent: (a) f is simple; (b) f is of finite deformation type; and (c) f is of finite Cohen-Macaulay type. - In particular a complete list of the normal forms for simple singularities is given, and, although the calculations in order to obtain the normal form of a given hypersurface are not completely described, a useful list of the main subcases is included. Also the adjacencies between the different types of simple singularities are completely given except for the case of surface singularities in characteristic 2 in which only partial information appears [see also \textit{F. Knop}, Invent. Math. 90, 579-604 (1987; Zbl 0648.14002)]. The characterizations of the singularities \(A_{\infty}, D_{\infty}\) are also obtained by the same methods. The differences to the well known case of characteristic 0 are adequately explained and some useful examples are given in order to make clear the exceptions appearing only in positive characteristic. ADE singularities; hypersurface singularities of finite deformation type; hypersurface singularities with finite Cohen-Macaulay type; normal forms for simple singularities Greuel, G.-M., Kröning, H.: Simple singularities in positive characteristic. Math. Z. 203(2), 339-354 (1990). Zbl 0715.14001 Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Local deformation theory, Artin approximation, etc., Local ground fields in algebraic geometry, Complex surface and hypersurface singularities Simple singularities in positive characteristic | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this paper the relationship between the classical description of the resolution of quotient singularities and the string picture is discussed in the context of \(N=(2,2)\) superconformal field theories. A method for the analysis of quotients locally of the form \(\mathbb{C}^d/G\) where \(G\) is abelian is presented. Methods derived from mirror symmetry are used to study the moduli space of the blowing-up process. The case \(\mathbb{C}^2/ \mathbb{Z}_n\) is analyzed explicitly. string theory; Calabi-Yau manifold; quantum geometry; resolution of quotient singularities; mirror symmetry P.S. Aspinwall, \textit{Resolution of orbifold singularities in string theory}, in \textit{Mirror symmetry II}, B. Greene and S.-T. Yau eds., American Mathematical Society (1996), pp. 355-379 [hep-th/9403123] [INSPIRE]. Global theory and resolution of singularities (algebro-geometric aspects), String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Modifications; resolution of singularities (complex-analytic aspects), Two-dimensional field theories, conformal field theories, etc. in quantum mechanics, Calabi-Yau manifolds (algebro-geometric aspects) Resolution of orbifold singularities in string theory | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(f\in \mathbb{C}[x_{1},x_{2},x_{3}]\) an \(ADE\) simple singularities and \(f^{T}\in \mathbb{C}[x_{1},x_{2},x_{3}]\) the corresponding Berglund-Hübsch dual. To construct a matrix model for the Fan-Jarvis-Ruan-Witten (FJRW) invariant of \(f^{T}\), similar to Kontesevich's model as in [\textit{M. Kontsevich}, Commun. Math. Phys. 147, No. 1, 1--23 (1992; Zbl 0756.35081)], one needs to find explicit identification of the generating function of FJRW invariants of \(f^{T}\) with a tau-function of a specific Kac-Wakimoto hierarchy. The identification involves rescaling parameters of the Kac-Wakimoto hierarchy and the precise value for the rescaling constants. Such a computation seems to be straightforward and it is not done in the paper. Instead, the authors explain in the introduction how the technical details leads to a problem in singularity theory.
On the other hand, in [\textit{H. Fan} et al., Ann. Math. (2) 178, No. 1, 1--106 (2013; Zbl 1310.32032)] the authors showed that generating functions of \(FJRW\) invariants of \(f^{T}\) coincides with the total descendant potentials of \(f\) (or, t.d.p. of \(f\), for short). In [\textit{E. Frenkel} et al., Funct. Anal. Other Math. 3, No. 1, 47--63 (2010; Zbl 1203.37108)] and [\textit{A. B. Givental} and \textit{T. E. Milanov}, Prog. Math. 232, 173--235 (2005; Zbl 1075.37025)] have proved that if \(f\) is a \(ADE\) singularity then t.d.p of \(f\) is a tau-function of the principal Kac-Wakimoto hierarchy of the same type \(ADE\). However, the problem is not solved yet, because the identification with the Milnor ring of the singularity and the Cartan subalgebra of the simple Lie algebra is not clearly explicit yet.
The contribution of the paper under review comes in showing that. The proofs are very well developed and they were done by approaching the ADE-singularity case by case. The paper offers a very pleasant reading. simple singularities; period map; mirror symmetry; topological K-theory Structure of families (Picard-Lefschetz, monodromy, etc.), Deformations of complex singularities; vanishing cycles, Equivariant \(K\)-theory Integral structure for simple singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(\Delta\) be a Euclidean quiver. We prove that the closures of the maximal orbits in the varieties of representations of \(\Delta\) are normal and Cohen-Macaulay (even complete intersections). Moreover, we give a generalization of this result for the tame concealed-canonical algebras.
Let \(A\) be a finite-dimensional \(k\)-algebra. Given a non-negative integer \(d\) one defines \(\text{mod}_A(d)\) as the set of all \(k\)-algebra homomorphisms from \(A\) to the algebra \(\mathbb M_{d\times d}(k)\) of \(d\times d\)-matrices. This set has a structure of an affine variety and its points represent \(d\)-dimensional \(A\)-modules. Consequently, we call \(\text{mod}_A(d)\) the variety of \(A\)-modules of dimension \(d\). The general linear group \(\text{GL}(d)\) acts on \(\text{mod}_A(d)\) by conjugation: \((g\cdot m)(a):=gm(a)g^{-1}\) for \(g\in\text{GL}(d)\), \(m\in\text{mod}_A(d)\) and \(a\in A\). The orbits with respect to this action are in one-to-one correspondence with the isomorphism classes of the \(d\)-dimensional \(A\)-modules. Given a \(d\)-dimensional \(A\)-module \(M\) we denote the orbit in \(\text{mod}_A(d)\) corresponding to the isomorphism class of \(M\) by \(\mathcal O(M)\) and its Zariski-closure by \(\overline{\mathcal O(M)}\).
The following theorem is the main result of the paper.
Theorem 1. Let \(M\) be a module over a tame hereditary algebra. If \(\mathcal O(M)\) is maximal, then \(\overline{\mathcal O(M)}\) is a normal complete intersection (in particular, Cohen-Macaulay).
Corollary 2. If \(M\) is an indecomposable module over a tame hereditary algebra, then \(\overline{\mathcal O(M)}\) is a normal complete intersection (in particular, Cohen-Macaulay).
Theorem 3. Let \(M\) be a \(\tau\)-periodic module over a tame hereditary algebra. If \(\mathcal O(M)\) is maximal, then \(\overline{\mathcal O(M)}\) is a complete intersection (in particular, Cohen-Macaulay).
We have the following generalization of Theorem 3.
Theorem 4. Let \(M\) be a \(\tau\)-periodic module over a tame concealed-canonical algebra such that \(\mathcal O(M)\) is maximal. Then \(\overline{\mathcal O(M)}\) is a complete intersection (in particular, Cohen-Macaulay). Moreover, \(\overline{\mathcal O(M)}\) is not normal if and only if \(\dim M\) is singular and \(\tau M\simeq M\).
The paper is organized as follows. In Section 1 we recall basic information about quivers and their representations. Next, in Section 2 we gather facts about the categories of modules over the tame concealed-canonical algebras. In Section 3 we introduce varieties of representations of quivers, while in Section 4 we review facts on semi-invariants with particular emphasis on the case of tame concealed-canonical algebras. Next, in Section 5 we present a series of facts, which we later use in Sections 6 and 7 to study orbit closures for the non-singular and singular dimension vectors, respectively. Moreover, in Section 7 we make a remark about relationship between the degenerations and the hom-order for the tame concealed-canonical algebras. Finally, in Section 8 we give the proof of Theorem 4. normal varieties; orbit closures; complete intersections; varieties of modules; Euclidean quivers; tame concealed-canonical algebras; varieties of representations; tame hereditary algebras; quiver representations Bobiński, G.: Normality of maximal orbit closures for Euclidean quivers, Can. J. Math. 64, No. 6, 1222-1247 (2012) Representations of quivers and partially ordered sets, Group actions on varieties or schemes (quotients), Representation type (finite, tame, wild, etc.) of associative algebras, Toric varieties, Newton polyhedra, Okounkov bodies, Group actions on affine varieties, Cohen-Macaulay modules in associative algebras Normality of maximal orbit closures for Euclidean quivers. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In the previous work [Ann. Glob. Anal. Geom. 35, No.~1, 91--114 (2009; Zbl 1188.53052)] the author develops a method of simultaneous desingularization of 3-dimensional singular special Lagrangian (SL) submanifolds with conical singularities of Calabi-Yau (CY) 3-folds with conical singularities at the same points, using methods of gluing constructions.
In this paper explicit examples are constructed, namely when the ambient space is the toroidal CY 3-orbifold \(T^6/\mathbb Z_3\) glued at each singular point with an asymptotically conical (AC) CY 3-fold a total space of the canonical line bundle \(K_{\mathbb{CP}^2}\), or when the ambient space is a certain quintic \(3\)-fold in \(\mathbb{CP}^4\) with ordinary double points glued with the cotangent bundle of \(\mathbb{S}^3\) as AC CY space (these spaces are examples of the constructions described by the author in [J. Math. 57, No. 2, 151--181 (2006; Zbl 1110.32010) and J. Math. 60, No. 1, 1--44 (2009; Zbl 1158.14302)]).
The SL 3-folds are compact and obtained by gluing AC SL 3-folds into singular SL 3-folds with conical singularities. These SL 3-folds are obtained using moment maps for some \(G\)-action in some CY manifold where \(G\) is a certain Lie group, or realized as the set of fixed points of an antiholomorphic involution. The AC SL 3-folds are in the corresponding AC CY 3-folds and the conically SL 3-folds in the corresponding conically singular CY 3-folds. Many examples of these SL submanifolds are described. Calabi-Yau manifolds; special Lagrangian submanifolds; conical singularities; asymptotically conical Chan, Y.-M.: Simultaneous desingularizations of Calabi--Yau and special Lagrangian 3-folds with conical singularities. II. Examples. Ann. Global Anal. Geom. (to appear) Calibrations and calibrated geometries, Calabi-Yau manifolds (algebro-geometric aspects), Lagrangian submanifolds; Maslov index Simultaneous desingularizations of Calabi-Yau and special Lagrangian 3-folds with conical singularities. II: Examples | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \(\mathfrak g\) be the simple Lie algebra of type \(G_2\) and \(C_i\) be the nilpotent conjugacy class in \(\mathfrak g\) of dimension \(i = 6, 8, 10\) and \(12\). In this paper the author shows the following:
(a) every conjugacy class of \(\mathfrak g\) except \(C_8\) has a normal closure with rational singularities,
(b) [\textit{T. Levasseur} and \textit{S. P. Smith}, J. Algebra 114, 81--105 (1988; Zbl 0644.17005)] \(\bar C_8\) is not normal in \(\bar C_6=\bar C_8\setminus C_8\). The normalization \(\eta_8: \tilde C_8\to \bar C_8\) is bijective and \(\tilde C_8\) has an isolated rational singularity in \(\eta_8^{-1}(0)\),
(c) \(\bar C_{12}\) has a singularity of type \(D_4\) in \(C_{10}\) and \(\bar C_{10}\) a singularity of type \(A_1\) in \(C_8\).
The proof of these results is based on the same construction as in Levasseur-Smith (loc. cit.). Namely, he embeds \(\mathfrak g\) into \(\mathfrak{so}_7\) by the 7-dimensional standard representation and studies the \(\mathfrak g\)-equivariant projection \(p: \mathfrak{so}_7\to \mathfrak g\).
At the end, the author gives a brief summary of what is known about normality of closures of conjugacy classes in reductive groups. maximal torus; Borel subalgebra; simple Lie algebra; nilpotent conjugacy class Kraft, H., Closures of conjugacy classes in \(G_2\), J. algebra, 126, 2, 454-465, (1989) Lie algebras of linear algebraic groups, Singularities in algebraic geometry, Group actions on varieties or schemes (quotients) Closures of conjugacy classes in \(G_2\) | 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 In this paper, the authors construct a natural complex structure on the space of maximal representations into any rank two real Lie group of Hermitian type. They describe the space of maximal components of the character variety of surface group representations into $\mathsf{PSp}(4,\mathbb{R})$ and $\mathsf{Sp}(4,\mathbb{R})$. Special attention is placed on components which contain singularities. To understand the singularities, the authors describe the local topology around every singular point and they show how the type of the singularity is related with the Zariski closure of the associated representation. They also describe the quotient of the maximal components of $\mathsf{PSp}(4,\mathbb{R})$ and $\mathsf{Sp}(4,\mathbb{R})$ by the action of the mapping class group as a holomorphic submersion over the moduli space of curves. For every real rank 2 Lie group of Hermitian type, the authors construct a mapping class group invariant complex structure on the maximal components. For the groups $\mathsf{PSp}(4,\mathbb{R})$ and $\mathsf{Sp}(4,\mathbb{R})$, they give a mapping class group invariant parametrization of each maximal component as an explicit holomorphic fiber bundle over Teichmüller space. Special attention is put on the connected components which are singular: the authors give a precise local description of the singularities and their geometric interpretation. They also describe the quotient of the maximal components of $\mathsf{PSp}(4,\mathbb{R})$ and $\mathsf{Sp}(4,\mathbb{R})$ by the action of the mapping class group as a holomorphic submersion over the moduli space of curves. These results are proven in two steps: first they use Higgs bundles to give a nonmapping class group equivariant parametrization, then they prove an analog of Labourie's conjecture for maximal $\mathsf{PSp}(4,\mathbb{R})$-representations. Let $\Gamma$ be the fundamental group of a closed oriented surface $S$ of genus at least two. The subspace of maximal representations of the $\mathsf{PSp}(4,\mathbb{R})$-character variety will be denoted by $\mathcal{X}^{\text{max}}\mathsf{PSp}(4,\mathbb{R})$. This paper is organized as follows: Section 1 is an introduction to the subject. In Section 2, the authors introduce character varieties, Higgs bundles and some Lie theory for the groups $\mathsf{PSp}(4,\mathbb{R})$ and $\mathsf{Sp}(4,\mathbb{R})$. In Section 3, they describe holomorphic orthogonal bundles, with special attention to the description of the moduli space of holomorphic $O(2,\mathbb{C})$-bundles. In Section 4, Higgs bundles over a fixed Riemann surface are used to describe the topology of $\mathcal{X}^{\text{max}}\mathsf{PSp}(4,\mathbb{R})$; special attention is placed on the singular components. In Section 5, they prove analogous results for $\mathcal{X}^{\text{max}}\mathsf{Sp}(4,\mathbb{R})$. In Section 6, they prove Labourie's conjecture concerning uniqueness of minimal surfaces. In Section 7, the universal Higgs bundle moduli space is constructed and in Section 8 the authors will put everything together and describe the action of $MCG(S)$ on $\mathcal{X}^{\text{max}}\mathsf{PSp}(4,\mathbb{R})$ and $\mathcal{X}^{\text{max}}\mathsf{PSp}(4,\mathbb{R})$. character varieties; mapping class group; Higgs bundles; maximal representations Discrete subgroups of Lie groups, Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills), Vector bundles on curves and their moduli, Fuchsian groups and their generalizations (group-theoretic aspects) The geometry of maximal components of the \(\mathsf{PSp}(4, \mathbb{R})\) character 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 This article studies symplectic algebraic geometry of Nakajima quiver varieties. These varieties give étale-local models of singularities, mostly along with a canonical symplectic resolution given by varying the stability parameter. The authors prove that Nakajima varieties are symplectic singularities in the sense of [\textit{A. Beauville}, Invent. Math. 139, No. 3, 541--549 (2000; Zbl 0958.14001)] (Theorem 1.2) and classify when they admit such a resolution, filling the lack for an explicit criterion. Their main result (Thm 1.5) states that for dimension vector \(\alpha \in \Sigma_{\lambda, \theta}\), the Nakajima quiver variety \(\mathfrak{M}_{\lambda}(\alpha, \theta)\) admits a projective symplectic resolution iff \(\alpha\) is indivisible or \((\gcd(\alpha), p(\gcd(\alpha)^{-1}\alpha)) = (2, 2)\).
In obtaining this result, the authors study the local and global algebraic symplectic geometry of quiver varieties, generalising Crawley-Boevey's decomposition theoreom [\textit{W. Crawley-Boevey}, Compos. Math. 130, No. 2, 225--239 (2002; Zbl 1031.16013)] and Le Bruyn's theorem computing the smooth locus [\textit{L. Le Bruyn}, J. Algebra 258, No. 1, 60--70 (2002; Zbl 1060.16015)], from affine quiver varieties to the non-affine case (Theorem 1.4 and Theorem 1.15, respectively). Further, the smoothness of the variety is characterised via the canonical decomposition (Corollary 1.17).
The paper is composed of seven sections. The introduction contains an overview of the main results and the second section provides a background on quiver varieties. The remaining five sections cover canonical decompositions of the quiver variety, the characterisation of the smooth points in terms of polystability (proof of Theorem 1.15 and of Corollary 1.17), the interesting \((2, 2)\) case from the main theorem, local factoriality of the quiver varieties and Namikawa's Weyl group. The ``technical heart of the paper'' is Section~6, discussing the local factoriality of the quiver variety. Here, the authors prove that for an indivisible anisotropic root not in the \((2,2)\) case, for a generic choice of the stability parameter \(\theta\), the quiver variety is locally factorial (Theorem 1.10). This section also contains the proofs of Theorems 1.2, 1.4 and 1.5. symplectic resolution; quiver variety; Poisson variety Representations of quivers and partially ordered sets, Poisson algebras, Geometric invariant theory, Symplectic structures of moduli spaces, Deformations of associative rings Symplectic resolutions of quiver varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0742.00082.]
The author describes the deformation theory of a simple singularity of type \(E_ 8\) from his point of view. The concept of the universal polynomial of type \(E_ 8\) plays the central role. It is a polynomial of degree 240 (= the number of root vectors in the root system of type \(E_ 8\)) with coefficients in the field of rational functions with 8 variables, and has the Galois group isomorphic to the Weyl group of type \(E_ 8\). Fix a point in the parameter space of the semi-universal deformation family of the simple singularity of type \(E_ 8\) and consider the fiber lying over it. Every fiber has a natural compactification \(S\). \(S\) has a structure of a rational elliptic surface with a section and with a singular fiber \(F\) of type II. \(F\) is isomorphic to the plane nodal cubic curve. Obviously the class \([F]\) of the singular fiber \(F\) in the Picard group \(\text{Pic}(S)\) of \(S\) satisfies \([F]^ 2=0\). The orthogonal complement \(L^*\) of \(L=\mathbb{Z}[F]\) contains \(L\) and the quotient module \(L^*/L\) with the induced bilinear form is isomorphic to the root lattice \(Q(E_ 8)\) of type \(E_ 8\) up to the signature of the bilinear forms. Thus the root system of \(L^*/L\), i.e., the collection of elements \(x\) in \(L^*/L\) with \(x^ 2=-2\), has 240 elements, which are called root vectors. The restriction morphism \(\text{Pic}(S)\to\text{Pic}(F)\) induces a morphism \(L^*/L\to\text{Pic}^ 0(F)\cong\mathbb{C}\) and we have 240 numbers in \(\mathbb{C}\) as the images of 240 root vectors. By definition the universal polynomial is the polynomial with these 240 numbers as roots. Its coefficients depend on the values of the parameters. The author shows that the every coefficient of this polynomial is a polynomial with rational coefficients of the parameters of the deformation family.
He told me that he would like to apply this concept to the non-abelian arithmetic theory of the field of rational numbers and the field of rational functions. However, this application is not developed in this article. universal polynomial of type \(E_ 8\); semi-universal deformation family of the simple singularity of type \(E_ 8\); universal polynoial of type \(E_ 8\) T. Shioda, Mordell-Weil lattices of type E 8 and deformation of singularities, in: Lecture Notes in Math. 1468 (1991), 177-202. Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry Mordell-Weil lattices of type \(E_ 8\) and deformation 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 Let \(G\) be a simple classical algebraic group over an algebraically closed field \(K\) of characteristic \(p\geq 0\). If \(G\) has natural module \(V\), one writes \(G=\text{Cl}(V)\). A maximal closed subgroup of \(G\) is called a subspace subgroup if it is reducible on \(V\), or if it is an orthogonal group on \(V\) embedded in a symplectic group with \(p=2\).
The author obtains the following result from which he then deduces several corollaries: Let \(G=\text{Cl}(V)\), let \(H\) be a maximal closed subgroup of \(G\) which is not a subspace subgroup, and let \(x\in G\) be a non-scalar semisimple or unipotent element of prime order. Then
\[
\tfrac{\dim(x^G\cap H)}{\dim x^G}\leq\tfrac 12+\varepsilon,
\]
where \(\varepsilon=0\) or \((G,H^0,\varepsilon)\) is given in the following list: (i) \(G=\text{SL}_{2n}\), \(p\) is arbitrary, \(H^0=\text{Sp}_{2n}\), \(\varepsilon=1/2n\); (ii) \(G=\text{Sp}_{2n}\), \(p\) is arbitrary, \(H^0=\text{Sp}_n^2\), \(\varepsilon=1/(2n+2)\); (iii) \(G=\text{SO}_{2n}\), \(p\) is arbitrary, \(H^0=\text{GL}_n\), \(\varepsilon=1/(2n-2)\); (iv) \(G=\text{SO}_7\), \(p\neq 2\), \(H^0=G_2\), \(\varepsilon=1/4\); (v) \(G=\text{Sp}_6\), \(p=2\), \(H^0=G_2\), \(\varepsilon=1/4\); (vi) \(G=\text{SO}_8\), \(p\neq 2\), \(H^0=\text{SO}_7\), \(\varepsilon=1/3\); (vii) \(G=\text{SO}_8\), \(p=2\), \(H^0=\text{Sp}_6\), \(\varepsilon=1/3\). classical algebraic groups; simple algebraic groups; homogeneous spaces; primitive permutation groups; fixed points; maximal closed subgroups; subspace subgroups T. C. Burness, Fixed point spaces in actions of classical algebraic groups, Journal of Group Theory 7 (2004), 311--346. Representation theory for linear algebraic groups, Group actions on varieties or schemes (quotients), Linear algebraic groups over arbitrary fields Fixed point spaces in actions of classical algebraic groups. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author proves a conjecture of Kontsevich on homotopy finiteness of certain DG categories, which states that the bounded derived category of coherent sheaves on a separated scheme of finite type over a field \(k\) of characteristic \(0\) is homotopically finitely presented, by establishing a smooth categorical compactification. The basic tools in the proof are categorical resolution of singularities and the closedness of admissible subcategories of smooth proper variety under gluing via perfect bimodules. A similar result for matrix factorizations is also established in the context of \(\mathbb{Z}/2\) graded DG categories.
The contents in more detail:
In section 1 the author gives an overview to the main content of the paper, briefly reviews some standard concepts in noncommutative algebraic geometry, states the main theorem and discuss its motivations.
In section 2 the author presents fundamental results about homotopy finiteness of DG categories and smooth categorical compactifications. Here the key concept is homotopical finite presentation (hfp) of small DG categories, which closely resembles that of finitely dominated spaces and is preserved under Drinfeld DG quotient construction.
In section 3 the author defines the notion of homological epimorphism of DG categories which generalize localization functors for small categories.
In section 4 and 5 the author discusses the gluing of DG categories via bimodules, and studies the properties of categories and functors under gluing. In particular, the gluing operation is compatible with localization and preserves hfp property.
In section 6 the author recall the notion of coderived category, which is the ind-completion of the complexes with bounded Noetherian cohomology, and the absolute derived category, which is the correct version of derived category for matrix factorization. These are backgrounds necessary for the extension of the main theorem to matrix factorization.
In section 7 the author constructs specific convenient enhancements for the category involved, and prove the versions of classical theorems for this enhancement.
The main theorem is finally proved in section 8, which roughly goes as follows: take a compactification \(Y\) of the variety we begin with, resolve its singularity by a sequence of blow up along smooth centers. By induction on the number of blow ups, the author constructs a DG category glued from smooth projective varieties with a localization functor to the derived category of \(Y\), hence establishes the main theorem. In the base step of the induction the author deals with derived categories of varieties whose reduced part is smooth by a Auslander-type construction, embedding \(D^b_{coh}(Y)\) into a compactification obtained by gluing copies of \(D^b_{coh}(Y_{red})\), and the induction steps are proved by a categorical blow construction which partially compactify the derived category of a variety by gluing (the derived categories of) its blow up and the center. A similar result for matrix factorizations follows the same lines. derived categories; differential graded categories; homotopy finiteness; Verdier localization; resolution of singularities Fundamental constructions in algebraic geometry involving higher and derived categories (homotopical algebraic geometry, derived algebraic geometry, etc.), Rational and birational maps, Localization of categories, calculus of fractions, Derived categories, triangulated categories Homotopy finiteness of some DG categories from algebraic geometry | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This comprehensive memoir provides a profound contribution to the currently developing topic of homotopical and higher categorical structures in algebraic geometry. Its main purpose is to generalize the concept of affine schemes to an adequate construction in homotopical algebraic geometry, namely to the concept of affine stacks, and to show how these new objects can be used to treat several questions in rational and \(p\)-adic homotopy theory from a novel point of view.
One of the motivating ideas of the present work can be traced back to A. Grothendieck's monumental, visionary manuscript ``Pursuing Stacks'' (unpublished) from the early 1980s. In this famous program, Grothendieck sketched a problem which he called the ``schematization problem for homotopy types'' stating that for any affine scheme \(\text{Spec\,}A\) there should be an appropriate notion of ``\(\infty\)-stack in groupoids over \(\text{Spec\,}A\)'' generalizing all sheaves and stacks in groupoids over the grand fpqc-site of \(\text{Spec\,}A\). Later on, it was shown by \textit{A. Joyal} (Letter to Grothendieck, 1984, unpublished) and by \textit{J. F. Jardine} [Homology Homotopy Appl. 3, No. 2, 361--384 (2001; Zbl 0995.18006)] that an adequate model for a theory of \(\infty\)-stacks in groupoids would be given by the theory of simplicial presheaves. According to this fundamental insight, the word ``stack'' in the present memoir, is used for an object in the homotopical category of simplicial sheaves (à la A. Joyal and J. F. Jardine).
After a thorough introduction to the contents of the present treatise, including a historical sketch of the developments leading to its subject, explanations of the basic conceptual framework, and an, outline of the links to related works by other researchers in the field. Section 1 recalls the fundamentals from the theory of simplicial presheaves on a Grothendieck site, that is from the theory of general stacks in the sense made precise above. Apart from an introduction to the basic definitions and results, homotopic limits, Postnikov decompositions, the cohomology of simplicial presheaves, and schemes in affine groups are the main topics of this preparatory section.
Section 2 introduces the first of the two novel fundamental notions of the memoir under review: affine stacks. These objects appear as a homotopic version of ordinary affine schemes, obtained from a model category with simplicial structure over the category of co-simplicial \(A\)-algebras. This model category is then used to define a derived functor of the Spec-functor, which in turn induces a certain functor from the homotopical category of co-simplicial \(A\)-algebras to the homotopical category of simplicial sheaves over the site \((\text{Aff}/A)_{\text{fpgc}}\). The category of affine stacks is then defined to be the essential image category of the latter functor. In the sequel, it is proved that the category of affine stacks is equivalent to the opposite category of the homotopical category of co-simplicial \(A\)-algebras, thereby generalizing the analoguous property of the category of ordinary affine schemes.
Finally, the author invents another important construction, namely that of the ``affinization of a simplicial presheaf'', which appears to be significant for the study of homotopy sheaves, and he gives a concrete criterion for the existence of affinizations. In the special case of a base scheme \(\text{Spec\,}k\), where \(k\) is a field, the affine stacks are completely characterized, and analogues of the standard theorems on rational and \(p\)-adic homotopy of algebraic varieties are deduced by using affine stacks.
Section 3 deals with the second crucial novelty provided by the author's work. More precisely, he introduces the notion of so-called ``affine \(\infty\)-gerbes'' and the concept of ``schematic homotopy type''. The underlying idea is to use affine stacks in order to define a homotopical version of affine gerbes in the Tannakian formalism (à la P. Deligne). This is done by glueing affine stacks to obtain so-called ``\(\infty\)-geometric stacks'', which may be seen as a generalization of ordinary algebraic stacks assed in algebraic moduli theory and non-abelian Hodge theory (à la C. Simpson). This framework is then applied to give two different solutions to A. Grothendieck's ``schematization problem for homotopy types of algebraic varieties'' mentioned above. In this context, homotopy types with respect to various cohomology theories (Betti, de Rham, crystalline, \(\ell\)-adic, etc.) are described in greater detail.
The concluding Section 4 is exclusively devoted to the study of ``\(\infty\)-geometric stacks'' over a ground field \(k\), with applications to the construction of analogues of some moduli stacks in this extended homotopical context.
In an appendix of the present memoir, the author recalls A. Grothendieck's ``schematization problem for homotopy types'', together with his interpretation and reflection of Grothendieck's vague sketches.
Without any doubt, this memoir is of utmost fundamental and propelling character with regard to further developments in homotopical algebraic geometry. A wealth of significant new concepts, methods, and applications is presented in a very detailed and lucid manner, and an important new point of view towards Grothendieck's program of pursuing stacks is strikingly exhibited. homotopical algebraic geometry; gerbes; Grothendieck topologies; cohomology theories; simplicial sheaves; model categories Toën, Bertrand, Champs affines, Selecta Math. (N.S.), 1022-1824, 12, 1, 39-135, (2006) Generalizations (algebraic spaces, stacks), Classical real and complex (co)homology in algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry, Étale and other Grothendieck topologies and (co)homologies, Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects), Grothendieck topologies and Grothendieck topoi, Topoi Affine stacks | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a variety. The triangulated category \(D^b(X)\) of bounded complexes contains the full triangulated subcategory \(Perf(X)\) of perfect complexes. If \(X\) is smooth, then \(Perf(X) = D^b(X)\), but this is no longer true if \(X\) is singular. \textit{D. Orlov} [``Triangulated categories of singularities and D-branes in Landau-Ginzburg models'', \url{arXiv:math/0302304}] defined the derived category of singularities \(D_{sg}(X)\) of \(X\) as the quotient triangulated category of \(D^b(X)\) by \(Perf(X)\). This category turns out to be very important for the study of mirror symmetry for Fano varieties.
In this paper, the authors consider \(\mathcal X\) and \(\mathcal Y\) two smooth Deligne-Mumford stacks with a pair of functors \(f: {\mathcal X} \to {\mathbb A}^1\) and \(g: {\mathcal Y} \to {\mathbb A}^1\). Assuming that there exists a relative Fourier--Mukai equivalence between \(D^b({\mathcal X})\) and \(D^b({\mathcal Y})\) whose kernel is a complex of sheaves on the fibered product \({\mathcal X} \times_{{\mathbb A}^1} {\mathcal Y}\), they show that the induced Fourier-Mukai functor between the derived categories of singularities of the fibers of \(f\) and \(g\) over \(0\) is also an equivalence.
The second part of the paper is dedicated to applications of this result in the settings of the McKay correspondence and toric geometry. derived category; singular category; Landau-Ginzburg model; McKay correspondence doi:10.2478/s11533-009-0063-y Singularities in algebraic geometry, Calabi-Yau manifolds (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies On equivalences of derived and singular 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 There is a well-known relation between simple algebraic groups and simple singularities. Simple singularities appear as generic singularities in codimension two of the unipotent variety of simple algebraic groups. The semi-universal deformation and the simultaneous resolution of the singularity can be constructed in terms of the algebraic group.
The present paper extends this relation to loop groups and simple elliptic singularities. It is a successful completion of the work of the second author for more general situations, and the paper (finished by the first author after the sudden death of the second one) is the final version of previous articles in which some of the problems had not yet been resolved. simple elliptic singularities; deformations Brieskorn, E.: Zur differentialtopologischen und analytischen Klassifikation gewisser algebraische Mannigfaltigkeiten. Dissertation, Bonn (1962) Deformations of singularities, Elliptic curves, Vector bundles on curves and their moduli, Complex surface and hypersurface singularities, Group actions on varieties or schemes (quotients) Loop groups, elliptic singularities and principal bundles over elliptic curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper sets out to generalize the results of \textit{A. Lubotzky} and \textit{A. Magid} [in Mem. Am. Math. Soc. 336 (1985; Zbl 0598.14042)] from nilpotent to solvable groups. In that paper a construction is described assigning to a finitely generated group \(\Gamma\) an algebraic variety \(S_ n(\Gamma)\) parametrizing equivalence classes of irreducible representations \(\rho\) : \(\Gamma\to GL_ n(k)\), where k is an algebraically closed field of characteristic zero. Lubotzky and Magid prove that for \(\Gamma\) nilpotent, \(S_ n(\Gamma)\) is a finite union of `twist-classes':
\[
S_ n(\Gamma)=\cup_{i}C_{\tau}(\rho_ i)\text{ with } C_{\tau}(\rho)=\{\lambda \otimes \rho | \quad \lambda \in Hom(\Gamma,GL_ 1(k)\}.
\]
Moreover, the representations \(\rho_ i\) may be taken to have finite image and these `twist-classes' are open, non- singular and of dimension \(rk(\Gamma^{ab})\)- the torsion-free rank of \(\Gamma\) made Abelian. For solvable groups all this is no longer true; however, we do prove Theorem 2.4, which asserts roughly that for \(\Gamma\) solvable, \(S_ n(\Gamma)\) is a finite union of `induced-twist classes'. While for \(\Gamma\) nilpotent \(S_ n(\Gamma)\) is non-singular, we give an example of a solvable group \(\Gamma\) for which \(S_ n(\Gamma)\) has singularities. This answers a question posed in [loc. cit.] and in fact is the first example known to the author of singularities in \(S_ n(\Gamma)\). solvable groups; finitely generated group; algebraic variety; irreducible representations; singularities [2] Rudnick Z., ''Representation varieties of solvable groups'', J. Pure and Applied Algebra, 45 (1987), 261--272 Representation theory for linear algebraic groups, Solvable groups, supersolvable groups, Classical groups (algebro-geometric aspects), Ordinary representations and characters, Group actions on varieties or schemes (quotients) Representation varieties of solvable 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 thesis is organized as follows. In Section 1 we give a quick review of the definition of Kleinian singularities. In Section 2 we introduce the deformation and resolution theory of singularities. In Section 3 we collect the basic notations and definitions of the representation theory of quivers. In Section 4 we present the main results from the deformation and resolution theory of the Kleinian singularities which are related to quiver varieties. These results are given by Kronheimer, Cassens and Slodowy. In Section 5 we speak about the nilpotent and stable representations of quivers and related results. These results are given by Lusztig and Hille. We explain how these results apply to a description of the exceptional set of the minimal resolution of a Kleinian singularity. In Section 6 we describe explicitly the intersection diagram \(\Gamma(\widetilde\mathbb{A}_{n-1})\) and consider action of the Weyl group on the space of weights \(\mathbb{H}(\delta)\). In Section 7 we describe the intersection diagrams \(\Gamma(\widetilde\mathbb{D}_4)\) and \(\Gamma(\widetilde\mathbb{D}_5)\). Kleinian singularities; deformations; resolutions; representations of quivers; quiver varieties; intersection diagrams; weights Representations of quivers and partially ordered sets, Deformations of singularities, Singularities in algebraic geometry, Deformations of associative rings McKay quivers and the deformation and resolution theory of Kleinian singularities. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(T\) be a maximal torus of the group \(\text{SL}(n)\). It was proved by \textit{J. Morand} [C. R. Acad. Sci., Paris, Sér. I, Math. 328, No. 3, 197-202 (1999; Zbl 0970.20027)] that the closure of each torus orbit in the adjoint representation is a normal variety. In this paper the author classifies all simple rational \(\text{SL}(n)\)-modules \(V\) with this property, i.e. for each point \(v\in V\) the closure \(\overline{Tv}\) is a normal affine variety (*).
The main theorem states that the only irreducible representations that satisfy this condition are: (1) the tautological representation; (2) the adjoint representation; (3) seven exceptional cases; and representations dual to these. -- The exceptional cases appear for \(n\leq 6\).
To prove that these representations satisfy (*) the author uses combinatorial arguments, some from graph theory. To construct nonnormal closures for other representations the author proceeds in two steps. First the case of fundamental representations is solved. This is done by reducing all possibilities (depending on \(n\) and the weight vector) to a few special cases. Then the author shows how to deal with all other weights.
The article is well-written and can be read by any person with basic knowledge on representation theory. Although many of the arguments rely on combinatorics, all notions are either explained or are commonly known. maximal tori; toric varieties; normality; saturation; orbits; simple rational modules; irreducible representations K. Kuyumzhiyan, ''Simple SL(n)-modules with normal closures of maximal torus orbits,'' J. Algebraic Combin. 30(4), 515--538 (2009). Representation theory for linear algebraic groups, Group actions on varieties or schemes (quotients), Homogeneous spaces and generalizations, Toric varieties, Newton polyhedra, Okounkov bodies, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) Simple \(\text{SL}(n)\)-modules with normal closures of maximal torus 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 The book under review presents the elements of the singularity theory of analytic spaces with applications; it consists of a preface, two main parts, three appendices and a bibliography including 158 items among which are 18 references on works written by the authors with collaborators.
The first part deals with complex spaces and germs. It contains basic notions and results of the general theory such as the Weierstraß preparation theorem, the finite coherence theorem with applications, finite and flat morphisms, normalization, singular locus and relations with differential calculus. In addition, the cases of isolated hypersurface and plane curve singularities are treated. Thus the authors describe some well-known invariants of hypersurface singularities including the Milnor and Tjurina numbers and methods of their computation. They also discuss the concept of finite determinacy, the property of quasihomogeneity, algebraic group actions, the classification of simple singularities, the parameterization and resolution of plane curve singularities, the intersection multiplicity and the semigroup of values associated with a plane curve singularity, the conductor and other classical topological and analytic invariants.
The second part is concerned with local deformation theory of complex space germs. First the authors describe the general deformation theory of isolated singularities of complex spaces. Then the notions of versality, infinitesimal deformations and obstructions are considered in detail. The final section contains a new treatment of equisingular deformations of plane curve singularities including a proof for the smoothness of the \(\mu\)-constant stratum which is based on properties of deformations of the parametrization. This result is obtained, in fact, as a further development of ideas by \textit{J. M. Wahl} [Trans. Am. Math. Soc. 193, 143--170 (1974; Zbl 0294.14007)]. Three appendices include a detail description of basic notions and results from sheaf theory, commutative algebra and formal deformation theory.
The book is written in a clear style, almost all key topics are followed by carefully chosen computational examples together with algorithms implemented in the computer algebra system \textsl{Singular} [see \textit{G.-M. Greuel, G. Pfister} and \textit{H. Schönemann}, Singular 3. A computer algebra system for polynomial computations. Centre for Computer Algebra, Univ. Kaiserslautern (2005), \url{http://www.singular.uni-kl.de}]. Moreover, the exposition contains many non-formal comments, remarks and good exercises illustrated by nice pictures. Without a doubt this book is comprehensible, interesting and useful for graduate students; it is also very valuable for advanced researchers, lecturers, and practicians working in singularity theory, algebraic geometry, complex analysis, commutative algebra, topology, and in other fields of pure mathematics. complex spaces and germs; isolated hypersurface singularities; equisingular deformations; \(\mu\)-constant stratum; embedded deformations; plane curve singularities; parametrization; resolution; normalization; versal deformations; obstructions; cotangent complex G.-M. Greuel, C. Lossen, E. Shustin, \(Introduction to Singularities and Deformations\) (Springer, Berlin, 2007) Deformations of complex singularities; vanishing cycles, Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces, Invariants of analytic local rings, Equisingularity (topological and analytic), Complex surface and hypersurface singularities, Modifications; resolution of singularities (complex-analytic aspects), Deformations of singularities, Singularities of surfaces or higher-dimensional varieties, Global theory and resolution of singularities (algebro-geometric aspects), Research exposition (monographs, survey articles) pertaining to algebraic geometry Introduction to singularities and deformations | 0 |
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