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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 There is a natural map \(\Delta\) from the moduli space \({\mathcal M}_{\mathcal O}\) of semistable rank two vector bundles with trivial determinant on a smooth projective curve \(C\) into the linear system \(|2\theta|\), where \(\theta\) is the theta divisor of the Jacobian of \(C\). According to results of Narasimhan and Ramanan, Beauville, Laszlo and Brivo and Verra, \(\Delta\) is an isomorphism if \(g(C)=2\), an embedding, if either \(g(C)=3\) and \(C\) is non hyperelliptic, or if \(g(C)>3\) and \(C\) is general. In the present paper it is shown that \(\Delta\) is an embedding for all non hyperelliptic curves \(C\) of genus \(g(C) >3\). Moreover its tangent spaces at singular points are described. semi-stable vector bundle; theta-divisor of Jacobian Van Geemen, B.; Izadi, E., \textit{the tangent space to the moduli space of vector bundles on a curve and the singular locus of the theta divisor of the Jacobian}, J. Algebraic Geom., 10, 133-177, (2001) Vector bundles on curves and their moduli, Jacobians, Prym varieties, Theta functions and abelian varieties, Families, moduli of curves (algebraic), Algebraic moduli problems, moduli of vector bundles The tangent space to the moduli space of vector bundles on a curve and the singular locus on the theta divisor of the Jacobian | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \((X, X^!)\) be a pair of Nakajima quiver varieties dual to each other under 3d mirror symmetry. Following results in [\textit{M. Aganagic} and \textit{A. Okounkov}, J. Am. Math. Soc. 34, No. 1, 79--133 (2021; Zbl 07304878)], a general enumerative expectation first proposed by Okounkov is that, up to the exchange of equivariant and Kähler variables prescribed by 3d mirror symmetry,
\[
\mathsf{V}_{\mathrm{QM}}(X^!) = \mathrm{Stab}^{\mathsf{Ell}}(X) \mathsf{V}_{\mathrm{QM}}(X)
\]
where \(\mathsf{V}_{\mathrm{QM}}(X)\) is the equivariant K-theoretic quasimap vertex of \(X\) and \(\mathrm{Stab}^{\mathsf{Ell}}(X)\) is a certain normalization of the elliptic stable envelope of \(X\). The important and simplest case, when \(X = T^*\mathrm{GL}(n)/B\) is the cotangent bundle of the full flag variety, is known to be 3d mirror to itself ([\textit{R. Rimányi} et al., SIGMA, Symmetry Integrability Geom. Methods Appl. 15, Paper 093, 22 p. (2019; Zbl 1451.53116)]) and the relation above was verified manually by Dinkins. The paper under review then takes the index limit ([\textit{N. Nekrasov} and \textit{A. Okounkov}, Algebr. Geom. 3, No. 3, 320--369 (2016; Zbl 1369.14069)]) of this relation, where:
\begin{itemize}
\item the right-hand side becomes K-theoretic stable envelopes of \(X\) or of certain subvarieties of \(X\) (see [\textit{Y. Kononov} and \textit{A. Smirnov}, Lett. Math. Phys. 112, No. 4, Paper No. 69, 25 p. (2022; Zbl 07569277)]);
\item the left-hand side becomes the so-called index vertex of \(X^!\), which can be viewed as a generalization of the refined topological vertex of Iqbal-Koczaz-Vafa.
\end{itemize}
This holds for index limits of any, not necessarily generic, slope. One consequence is that index vertices are always expansions of rational functions. The simplest example \(X = T^*\mathbb{P}^1\) is given explicitly. stable envelopes; 3d mirror symmetry; quasimaps; enumerative geometry Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Elliptic cohomology, Enumerative problems (combinatorial problems) in algebraic geometry, Combinatorial aspects of algebraic geometry Euler characteristic of stable envelopes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Mirror symmetry emerged as a somewhat mysterious duality between algebraic and symplectic geometry. The attempts to understand the origin of this duality resulted in the formulation of homological mirror symmetry conjectures. While these conjectures are rather general and quite abstract, in various specific situations they result in more explicit, and still non-trivial statements. In particular this is so in the analysis of mirror symmetry for simple elliptic singularities, which is the subject of this paper. In this case a conjectural isomorphism between three different Frobenius manifolds can be regarded as a prediction of the homological mirror symmetry. The Frobenius manifolds in question are constructed respectively from the theory of a weighted projective line, the invariant theory for an elliptic Weyl group, and the theory of primitive forms for a universal unfolding of a simple elliptic singularity. In particular, the isomorphism between these Frobenius manifolds explains certain surprising relations between isolated singularities, root systems and discrete groups.
An interesting class of theories of a weighted projective line is that of \(\mathbb{P}^1_{\alpha_1,\alpha_2,\alpha_3}\). It has been shown recently by \textit{P. Rossi} [Math. Ann. 348, No. 2, 265--287 (2010; Zbl 1235.14053)] that the isomorphism between corresponding Frobenius manifolds mentioned above holds for \(1/\alpha_1 + 1/\alpha_2 + 1/\alpha_3 >1\). In this paper the authors analyze what happens for \(1/\alpha_1 + 1/\alpha_2 + 1/\alpha_3 =1\); the authors consider in detail the case \(\alpha_1=\alpha_2=\alpha_3 =3\) and identify the corresponding Frobenius manifolds. As a consequence of this analysis, a geometric interpretation of the Fourier coefficients of a certain modular function considered by Saito is found. The authors conduct a similar analysis for the case of \(\mathbb{P}^1_{2,2,2,2}\), which however does not have a hypersurface singularity. In both these situations the genus zero and one Gromov-Witten potentials are computed and expressed by quasi-modular forms.
The analysis in the paper boils down, to much extent, to explicit computations of genus zero and genus one Gromov-Witten potentials in for \(\mathbb{P}^1_{3,3,3}\) and \(\mathbb{P}^1_{2,2,2,2}\). These results should be of interest to those working in the related topics: mirror symmetry, Frobenius manifolds, singularity theory, etc. They also provide a nice illustration of how deep ideas (in mirror symmetry) can be related to explicit computations. mirror symmetry; simple elliptic singularities; Frobenius manifolds; Gromov-Witten theory; eta product Basalaev, A.: \(\text{SL}(2, \mathbb{C})\)-action on cohomological field theories and Gromov-Witten theory of elliptic orbifolds. Oberwolfach reports, 22/2015 (2015) Mirror symmetry (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Complex surface and hypersurface singularities, Gromov-Witten invariants, quantum cohomology, Frobenius manifolds Gromov-Witten invariants for mirror orbifolds of 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 Let \(\Gamma\subset\text{SL}_2(\mathbb{C})\) be a finite nontrivial subgroup. Associated to \(\Gamma\) is a Kleinian singularity \(X:=\mathbb{C}^2/\Gamma\). \textit{W. Crawley-Boevey} and \textit{M. P. Holland} [Duke Math. J. 92, No. 3, 605-635 (1998; Zbl 0974.16007)] introduced a family \(\{\mathcal O^\lambda\}\) of (generally) noncommutative deformations of \(X\). The main result of this paper is the construction for certain \(\lambda\) with \(\mathcal O^\lambda\) having finite global dimension of a family of filtered \(\mathbb{Z}\)-algebras \(\{B^\lambda(\chi)\}\) with each \(B^\lambda(\chi)\) being Morita equivalent to \(\mathcal O^\lambda\). This is an analogue of a result of \textit{I. Gordon} and \textit{J. T. Stafford} [Adv. Math. 198, No. 1, 222-274 (2005; Zbl 1084.14005)] which states that certain rational Cherednik algebras are Morita equivalent to certain noncommutative deformations of a Hilbert scheme. Both of these results were conjectured by V. Ginzburg. This result provides a new approach to studying the representation theory of \(\mathcal O^\lambda\) which the author investigates in forthcoming work.
Moreover, the associated graded \(\mathbb{Z}\)-algebra of \(B^\lambda(\chi)\) is shown to be Morita equivalent to the minimal resolution of \(X\). This affirmatively answers a question posed by \textit{M. P. Holland} [Ann. Sci. Éc. Norm. Supér. (4) 32, No. 6, 813-834 (1999; Zbl 1036.16024)]. In the sense of Holland's work, the algebra \(\mathcal O^\lambda\) can be considered as a `quantization' of \(X\), and Holland asked whether the minimal resolution of \(X\) admitted a quantization. This result shows that these graded algebras are such quantizations.
The paper includes a discussion of quivers, particularly the McKay quiver of \(\Gamma\) which is needed for the definition of \(\mathcal O^\lambda\), as well as of the construction of minimal resolutions of \(X\). The author also discusses some fundamental notions about \(\mathbb{Z}\)-algebras and introduces the concept of a Morita \(\mathbb{Z}\)-algebra. A key step in the proof is a stronger version of a categorical equivalence result of Gordon and Stafford.
It should be noted that using independent methods a slightly different version of the main result here has been proved by \textit{I. M. Musson} [J. Algebra 293, No. 1, 102-129 (2005; Zbl 1082.14008)] for Kleinian singularities of type \(A\). Kleinian singularities; noncommutative deformations; minimal resolution; quantizations; \(\mathbb{Z}\)-algebras M. Boyarchenko, Quantization of minimal resolutions of Kleinian singularities, Adv. Math., 211 (2007), 244--265. Rings arising from noncommutative algebraic geometry, Noncommutative algebraic geometry, Deformations of singularities, Geometric invariant theory, Representations of associative Artinian rings, Module categories in associative algebras, Deformations of associative rings, Modifications; resolution of singularities (complex-analytic aspects), Graded rings and modules (associative rings and algebras) Quantization of minimal resolutions 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 smooth projective variety and \(D \subset X\) a divisor with simple normal crossings. The paper discusses a Deligne glueing for the Hitchin twistor space of the moduli of local systems over \(M=X \setminus D\).
After some preliminary definitions, the Deligne glueing in the compact case is recalled in section 3. Then the simplest non-compact case, where \(X=\mathbb{P}^1\) is the projective line and \(D=\{0,\infty\}\), is treated with full details in section 4. In particular the line bundle \(O_{\mathbb{P}^1}(2)\) occurs naturally in this situation with a new rich structure, called the Tate twistor structure, whose study is the aim of section 5.
In section 6 the author discusses the general rank one case. The twistor space of representations maps to a weight two space of local monodromy transformations around a component \(D_1\) of \(D\). The space of \(\sigma\)-invariant sections of this bundle is a real 3-dimensional space whose parameters correspond to the complex residue of the Higgs field along \(D_1\) and the real parabolic weight of a harmonic bundle.
The paper ends with a strictness conjecture for natural mappings between two moduli spaces of representations on two quasi-projective varieties \(M\) and \(M'\) as above. connection; fundamental group; representation; quasi-projective variety; Higgs bundle; twistor space Simpson, C.T.: A weight two phenomenon for the moduli of rank one local systems on open varieties. From Hodge theory to integrability and TQFT \textit{tt}*-geometry. In: Proceedings of Symposia in Pure Mathematics, vol. 78, pp. 175-214. American Mathematical Society, Providence, RI (2008) Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Transcendental methods of algebraic geometry (complex-analytic aspects), Transcendental methods, Hodge theory (algebro-geometric aspects), Homotopy theory and fundamental groups in algebraic geometry, Twistor theory, double fibrations (complex-analytic aspects) A weight two phenomenon for the moduli of rank one local systems on open varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The main object of the paper under review is the complex vector space \(Q_ n^ d\) of \(n\)-ary forms of degree \(d\) with complex coefficients (\(n\geq 2\), \(d\geq 3\)). More precisely, the authors are interested in the map \(\Phi\) sending a nondegenerate \(n\)-ary \(d\)-form \(f\) to the associated form which is an element of the space dual to \(Q_ n ^{n(d-2)}\). Their focus is on the cases of binary quartics and ternary cubics, the only cases when \(\Phi\) preserves the degree. They show that in each of these cases the projectivization of \(\Phi\) induces an equivariant (with respect to an action of \(SL_n\)) involution on the projectivization of the space of nondegenerate forms with one orbit removed. Furthermore, they show that a such a nontrivial rational equivariant involution is unique. In particular, \(\Phi\) yields a unique equivariant involution on the space of elliptic curves with nonvanishing \(j\)-invariant. They give a simple interpretation of this involution in terms of projective duality and express it via classical contravariants, in the spirit of Cayley and Sylvester.
Eventual applications in singularity theory are also explained. geometric invariant theory; binary quartic; ternary cubic; elliptic curve; contravariant J. Alper, A. Isaev and N. G. Kruzhilin, Associated forms of binary quartics and ternary cubics, Transform. Groups (2015), 10.1007/S00031-015-9343-8. Geometric invariant theory, General ternary and quaternary quadratic forms; forms of more than two variables, Elliptic curves, Invariants of analytic local rings, Complex surface and hypersurface singularities Associated forms of binary quartics and ternary cubics | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 has two parts. In the first, one studies for non-isolated hypersurface singularities questions as: finite determinacy, unfoldings and deformations, the topology of the nearby fibres. Best results are given when the singular locus has dimension one. The key point is to fix an analytic germ \(\Sigma\) in \(({\mathbb{C}}^ m,0)\) and to look to holomorphic functions which contain \(\Sigma\) in their singular locus and to coordinate transformations which leave \(\Sigma\) invariant. This leads to the study of the corresponding right-equivalence relation and of two algebraic notions: the primitive ideal associated to \(\Sigma\) and Jacobi modules. The primitive ideal of \(\Sigma\) is given exactly by the functions as above, i.e. it is \(\{f| (f)+J_ f\subset I_{\Sigma}\}\) \((J_ f\) is the Jacobi ideal of f). A Jacobi module is nothing else than a quotient \(I_{\Sigma}/J_ f.\)
In the second part (pure algebraic) one studies the depth and the projective dimension of the quotient of two ideals. This topic emerges naturally from the first part and is used in the first part. depth of a module; resolutions of Jacobi modules; non-isolated hypersurface singularities; unfoldings; deformations; Jacobi ideal; projective dimension Pellikaan, G. R.: Hypersurfaces singularities and resolutions of Jacobi modules. (1985) Local complex singularities, Complex singularities, Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces, Singularities of surfaces or higher-dimensional varieties, Deformations of complex singularities; vanishing cycles, Singularities in algebraic geometry Hypersurface singularities and resolutions of Jacobi modules | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(\mathbb{P}_{\Delta} \) be a toric Fano 4-variety with only Gorenstein singularities defined by a lattice reflexive polytope \(\Delta \subset \mathbb{R}^{4}\). A generic anti-canonical divisor \(Z_{\Delta}\) is a Calabi-Yau threefold with only canonical singularities, and a crepant resolution \(\overline{Z}_{\Delta}\) of \(Z_{\Delta}\) is a smooth Calabi-Yau threefold. \(\overline{Z}_{\Delta}\) is called a Batyrev Calabi-Yau threefold, since \textit{V. V. Batyrev} studied such Calabi-Yau threefolds in [J. Algebr. Geom. 3, No. 3, 493--535 (1994; Zbl 0829.14023)] for constructing hypothetical mirror pairs of Calabi-Yau manifolds. This paper proves that the automorphism group of a Batyrev Calabi-Yau threefold is finite. Let \(\overline{Z}_{\Delta}\) be a Batyrev Calabi-Yau threefold, \(K\subset H^{1,1}(\overline{Z}_{\Delta}, \mathbb{R})\) be the Kähler cone, i.e., \(\alpha \in K\) iff \(\alpha\) can be represented by a Kähler metric, and \(\overline{K}\) be the nef cone, i.e., the closure of \(K\) in \( H^{1,1}(\overline{Z}_{\Delta}, \mathbb{R})\). The second Chern class \(c_{2}(\overline{Z}_{\Delta})\) can be regarded as a linear functional on \(H^{1,1}(\overline{Z}_{\Delta}, \mathbb{R})\) by \(c_{2}(\overline{Z}_{\Delta})\cdot \alpha\), and it is positive on \(K\) by the Miyaoka-Yau inequality. In this paper, the author proves that \(c_{2}(\overline{Z}_{\Delta})\) is furthermore strictly positive on the nef cone \(\overline{K}\), and as consequence, obtains the finiteness of the automorphism group. If one grants the Morrison conjecture, i.e., for any Calabi-Yau threefold \(Z\), there is a rational polyhedral cone \(\Pi\subset H^{2}(Z, \mathbb{R})\) such that \(\overline{K}=\mathrm{Aut}(Z)\cdot \Pi\), then a Batyrev Calabi-Yau threefold has a polyhedral nef cone. Calabi-Yau manifolds Calabi-Yau manifolds (algebro-geometric aspects), Automorphisms of surfaces and higher-dimensional varieties Automorphism group of Batyrev Calabi-Yau threefolds | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 survey article deals with some classes of algebras arising in the representation theory of finite dimensional algebras which are related to exceptional curves. Roughly speaking, an exceptional curve is a possibly noncommutative curve which is noetherian, smooth, projective and admits an exceptional sequence of coherent sheaves. The concept of exceptional curves presented in this article generalizes the notion of weighted projective lines which was investigated by \textit{W. Geigle} and the author [in: Singularities, representations of algebras and vector bundles, Lect. Notes Math. 1273, 265-297 (1987; Zbl 0651.14006)]\ in order to study module categories for canonical algebras (see [\textit{C. M. Ringel}, Tame algebras and integral quadratic forms, Lect. Notes Math. 1099 (1984; Zbl 0546.16013)]) from a geometrical point of view. Whereas exceptional commutative spaces are quite rare, in fact the only exceptional commutative curve over an algebraically closed field is the projective line, there is a rich supply of noncommutative exceptional curves.
An exceptional curve \(\mathbb{X}\) is called homogeneous if for each point of \(\mathbb{X}\) there is only one simple sheaf which is concentrated in that point. It is shown that the geometry of homogeneous curves can be controlled completely by the representation theory of a finite dimensional tame bimodule algebra. In contrast to the commutative case it may happen that there exist more than one simple sheaf concentrated in a single point. The author describes how these curves arise from the homogeneous curves applying a process of insertion of weights and explains how this concept is related to that of vector bundles with parabolic structures in the sense of \textit{C. S. Seshadri} [Fibrés vectoriels sur les courbes algébriques, Astérisque 96 (1982; Zbl 0517.14008)].
Furthermore the author discusses many applications of the concept of exceptional curves. He shows that it applies besides to finite dimensional tame hereditary and canonical algebras also to coordinate algebras of surface singularities, preprojective algebras of tame hereditary algebras, algebras of automorphic forms and two-dimensional factorial algebras. For more details concerning these topics we refer to the papers of \textit{W. Geigle} and \textit{H. Lenzing} [J. Algebra 144, No. 2, 273-343 (1991; Zbl 0748.18007)], \textit{D. Baer, W. Geigle} and \textit{H. Lenzing} [Commun. Algebra 15, 425-457 (1987; Zbl 0612.16015)], \textit{H. Lenzing} [in: Finite dimensional algebras and related topics, 191-212 (1994; see the following review Zbl 0895.16004)]\ and \textit{D. Kussin} [Graded factorial algebras of dimension two, Bull. Lond. Math. Soc. (to appear)]. survey; finite dimensional algebras; exceptional curves; noncommutative curves; exceptional sequences of coherent sheaves; weighted projective lines; module categories for canonical algebras; homogeneous curves; finite dimensional tame bimodule algebras; vector bundles with parabolic structures; coordinate algebras; surface singularities; tame hereditary algebras H. Lenzing, Representations of finite dimensional algebras and singularity theory, \textit{Trends in ring theory} (Miskolc, Hungary, 1996), \textit{Canadian Math. Soc. Conf. Proc.,}\textbf{22} (1998), Am. Math. Soc., Providence, RI (1998), 71-97. Representations of quivers and partially ordered sets, Singularities of curves, local rings, Representation type (finite, tame, wild, etc.) of associative algebras, Elliptic curves, Vector bundles on curves and their moduli Representations of finite dimensional algebras and singularity theory | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(k\) be an algebraically closed field of characteristic \(p>0\) and \(X_1, X_2\) be two smooth proper connected curves. Let \(\sigma_i: X_i\to X_i\) be an automorphism of order \(p\) and denote by \(\sigma\) the automorphism \(\sigma_1\times \sigma_2: X_1\times X_2\to X_1\times X_2=:Y\). It is proved that the graph of the resolution of any singularity of \(Y/\langle\sigma \rangle\) is a star-shaped graph with three terminal chains when \(X_2\) is an ordinary curve of positive genus. The intersection matrix of the resolution has determinant \(\pm p^2\). The singularity is rational. It is proved that for any \(s>0\) not divisible by \(p\) there are resolution graphs of wild \(\mathbb{Z}/p\mathbb{Z}\) quotient singularities with one node, \(s+2\) terminal chains and intersection matrix having determinant \(\pm p^{s+1}\). product of curves; cyclic quotient singularity; rational singularity; wild; intersection matrix; resolution graph; fundamental cycle Lorenzini, D.: Wild quotients of products of curves (2012, preprint) Singularities in algebraic geometry, Local ground fields in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Singularities of curves, local rings, Multiplicity theory and related topics, Singularities of surfaces or higher-dimensional varieties Wild quotients of products of 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 studies crepant resolutions of quotient singularities of the form \(\mathbb C^3/G\), where \(G\) is a finite subgroup of \(\mathrm{SL}_3(\mathbb C)\). A crepant resolution of \(\mathbb C^3/G\) always exists, but unlike in the surface case, it is usually not unique. All crepant resolutions of \(\mathbb C^3/G\) are related by small \(\mathbb Q\)-factorial modifications.
The authors study and describe the Cox ring of a crepant resolution \( X_0 \to \mathbb C^3/ G\), and obtain information on the structure of the set of all crepant resolutions of \(\mathbb C^3/G\) such as their number, and relations between them. This information is encoded in the chamber decomposition of the movable cone \(\mathrm{Mov}(X_0)\) determined by the Cox ring. Based on previous work starting with [\textit{M. Donten-Bury} and \textit{J. A. Wiśniewski}, Kyoto J. Math. 57, No. 2, 395--434 (2017; Zbl 1390.14048)], the authors produce generating sets for the Cox ring of \(X_0\to \mathbb C^3/ G\) when \(G\) is a representation of a dihedral group, when \(G\) is a non-abelian reducible representation and in three cases where \(G\) is an irreducible representation. quotient singularity; resolution of singularities; crepant resolution; Cox ring Global theory and resolution of singularities (algebro-geometric aspects), Minimal model program (Mori theory, extremal rays), McKay correspondence, Group actions on varieties or schemes (quotients), Geometric invariant theory, Divisors, linear systems, invertible sheaves Crepant resolutions of 3-dimensional quotient singularities via Cox rings | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The concept of a Hitchin-Kobayashi correspondence plays a central role in connecting ideas of geometric invariant theory to those of gauge theory by relating a notion of stability to the solutions of a set of partial differential equations. The concept derives originally from work of \textit{M. S. Narasimhan} and \textit{C. S. Seshadri} [Ann. Math. (2) 82, 540--567 (1965; Zbl 0171.04803)] and was first introduced for holomorphic bundles. It has subsequently been developed by many authors for various types of augmented bundles, that is for bundles with some additional structure. A general framework intended to unify the Hitchin-Kobayashi correspondence for a large class of augmented bundles was introduced recently by \textit{I. Mundet i Riera} [J. Reine Angew. Math. 528, 41--80 (2000; Zbl 1002.53057)] building on ideas of \textit{D. Banfield} [Q. J. Math. 51, 417--436 (2000; Zbl 0979.53028)]. The key idea is that of a principal pair. This covers many types of augmented bundle, but there are important examples, including coherent systems and Higgs bundles, which do not fit the framework. In each of these cases, the problem is that the set of all augmented bundles of the given type corresponds to a proper subset of the corresponding principal pairs, and moreover the automorphism groups of the augmented bundles correspond to proper subgroups of the automorphism groups of the principal pairs.
The object of the present paper is to overcome this problem by replacing the gauge group of the principal bundle, which corresponds to the automorphism group of the principal pair, by a subgroup. The main difficulty is moving from the finite-dimensional setting of geometric invariant theory, where it is obvious how to replace the action of a reductive group by that of a reductive subgroup, to the infinite-dimensional gauge theoretic setting. Approximately half the paper is concerned with examples. These include tensor product bundles, triples in which the second bundle is fixed, coherent systems and Higgs bundles. geometric invariant theory; gauge theory; augmented bundle; stability; Hitchin-Kobayashi correspondence; coherent system; Higgs bundle; principal pair; moment map Bradlow, Steven B.; García-Prada, Oscar; Mundet i Riera, Ignasi, Relative Hitchin-Kobayashi correspondences for principal pairs, Q. J. Math., 54, 2, 171-208, (2003) Momentum maps; symplectic reduction, Algebraic moduli problems, moduli of vector bundles, Complex-analytic moduli problems Relative Hitchin-Kobayashi correspondences for principal pairs | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a nodal curve, i.e., \(X\) is reduced, possibly reducible, and has at most nodes as singularities. Let \(g\) be the arithmetic genus of \(X\). The compactified Picard variety \(\overline{P^{g-1}_X}\) of degree \(g-1\) is the finite union of irreducible \(g\)-dimensional components each of which contains as an open subset a copy of the generalized Jacobian of \(X\). It is known that the compactified Picard variety of any nodal curve has a polarization, the theta divisor \(\Theta(X)\). The author studies the theta divisor of the compactified Jacobian of a nodal curve \(X\). The first main result of the paper describes the irreducible components of \(\Theta(X)\) and establishes that every irreducible component of the compactified Jacobian contains a unique irreducible component of the theta divisor, unless \(X\) has some separating node. The proof uses the Abel map \(X^{g-1} \rightarrow \mathrm{Pic}^{g-1}\). It turns out that the theta divisor coincides with the closure of the image of the Abel map for every stable multidegree.
Another goal of the paper is the geometric interpretation of \(\Theta(X)\) and the precise description of the objets which are parametrized by \(\Theta(X)\). In particular, in this direction it is proved that a stratification of \(\overline{P^{g-1}_X}\) induces the canonical stratification on \(\Theta(X)\). This stratification of \(\Theta(X)\) allows to describe the partial normalizations of \(X\) in terms of effective line bundle.
The author applies the results and the techniques of the paper to generalize to singular curves the characterization of smooth hyperelliptic curves via the singular locus of their theta divisor. Nodal curve; line bundle; compactified Picard scheme; theta divisor; Abel map; hyperelliptic stable curve Caporaso L.: Geometry of the theta divisor of a compactified Jacobian. J. Eur. Math. Soc. 11, 1385--1427 (2009) Jacobians, Prym varieties, Special divisors on curves (gonality, Brill-Noether theory), Picard schemes, higher Jacobians, Algebraic moduli problems, moduli of vector bundles Geometry of the theta divisor of a compactified Jacobian | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper is about a formalization of known procedures to resolve singularities of algebraic varieties defined over fields of characteristic zero. This formalization is presented as a game, called \textit{Stratify}. Probably the name is due to the fact that, after all, a canonical resolution of singularities means to stratify, in a suitable natural way, the singular locus \(S\) of the variety (and those of its transforms) as a union of locally closed regular subvarieties, at each step of the desingularization process we blow up the ``worst'' stratum (which is closed).
At each stage of the game we have a finite weighted graph, which is successively modified by two players \(A\) and \(B\), according to certain rules. It is a rather curious game, because only player \(A\) can win (by reaching a \textit{final stage.} Player \(B\) can only prevent \(A\) from winning, perhaps forever. The game mimics, in a formal or combinatorial way, the different stages in an attempt to resolve singularities of varieties (in characteristic zero, where we may perform induction by using hypersurfaces of maximal contact), specially by taking blowups with permissible centers.
In the paper, after a review of known results on desingularization, the authors explain the game. The list of rules is rather long and complicated. Then the necessary algebro-geometric concepts are briefly reviewed. Next they explain how a resolution process can be translated into the rules of \textit{Stratify}. Actually, the process considered is not one trying to directly resolve singular algebraic varieties, but rather a resolution of pairs \((I,b)\), where \(I\) is a sheaf of nonzero ideals on a regular ambient variety and \(b\) is a natural number (or rather of some closely related objects, the \textit{singularity data}). It is known (and sketched in the paper) that solving this problem implies resolution of varieties. The authors prove that this resolution problem is equivalent to having a winning strategy for ``Stratify'', and finally they show that such a winning strategy is available. singularity; resolution; graph; transversality; singular datum; Rees algebra Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Polynomials in real and complex fields: location of zeros (algebraic theorems), Directed graphs (digraphs), tournaments A game for the 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 Let \(\mathbb{K}\) be an algebraically closed field of characteristic \(0\). Let \(G=\mathrm{SL}_n=\mathrm{SL}_n(\mathbb{K})\) and let \(B\) be a Borel subgroup of \(G\). Then \(G/B\) is isomorphic to the variety of complete flags in an \(n\)-dimensional vector space. The Plücker embedding realizes \(G/B\) inside the product of projective spaces of all fundamental representations of \(\mathrm{SL}_n\), which are isomorphic to the wedge powers of the vector representation in type \(A\). The coordinates \(X_I\) on the \(k\)-th fundamental representation are labeled by the cardinality \(k\) subsets of \(\{1,\dots,n\}\). The Plücker relations describe the image of this embedding, generating the ideal \(J_n\) of all multi-homogeneous polynomials vanishing on the image of the Plücker embedding since it is the ideal of relations satisfied by the minors of the matrices from \(\mathrm{SL}_n\). An important property of \(J_n\) is that the quotient of the polynomial ring \(R_n\) in variables \(X_I\) modulo the ideal \(J_n\) is isomorphic to the direct sum of dual irreducible finite-dimensional representations of \(\mathrm{SL}_n\).
The authors generalize this finite-dimensional setting to the semi-infinite one by replacing \(G\) with \(G[[t]]\) and representations \(V\) with infinite-dimensional spaces \(V[[t]]=V\otimes_{\mathbb{K}} \mathbb{K}[[t]]\). The primary geometric object \(\mathbf{Q}\) that the authors are interested in is the following: let \(V(\omega_k)\) be the fundamental irreducible modules of \(\mathrm{SL}_n\), where \(k=1,\ldots,n-1\). Let \(\mathring{\mathbf{Q}}\subset \displaystyle{\prod_{k=1}^{n-1}} \mathbb{P}(V(\omega_k)[[t]])\) be the \(G[[t]]\)-orbit through the product of highest weight lines, with the reduced scheme structure. The scheme \(\mathbb{Q}\) is defined as the Zariski closure of \(\mathring{\mathbb{Q}}\) inside the product of projective spaces.
The authors write down the semi-infinite Plücker relations, describing the Drinfeld-Plücker embedding of the semi-infinite flag varieties in type \(A\) (Section 3, page 4367). The authors then study the homogeneous coordinate ring, the quotient by the ideal generated by the semi-infinite Plücker relations. They establish an isomorphism with the algebra of dual global Weyl modules (Corollary 4.22, page 4391) and derive a new character formula (Corollary 4.21, page 4391). semi-infinite Plücker relations; Weyl modules; Drinfeld--Plücker embedding; semi-infinite flag varieties; Weyl modules Grassmannians, Schubert varieties, flag manifolds, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Simple, semisimple, reductive (super)algebras, Lie algebras of Lie groups, Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.) Semi-infinite Plücker relations and Weyl modules | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(\mathbb{S}_{d}\) be the \(\text{SL}_{2}( \mathbb{C})\)-module of degree \(d\) binary forms. Then \(\text{SL}_{2}( \mathbb{C} )\) acts on the cone of the Grassmannian of subspaces of a fixed dimension \(k\) in \(\mathbb{S}_{d},\) i.e. SL\(_{2}( \mathbb{C})\) acts on \(A_{k,d},\) the cone of \(\mathbb{G}_{k}( \mathbb{S}_{d}).\)
This paper examines the complication of the invariants of \(A_{k,d}^{\text{SL}_{2}( \mathbb{C})},\) the invariants combinants. This complication is denoted cpl \(A_{k,d}^{\text{SL}_{2}( \mathbb{C} )}.\)
There are two results presented here. The first is that, for any natural number \(n\) there are only finitely many pairs of integers \(k\) and \(d\) such that cpl \(A_{k,d}^{\text{SL}_{2}( \mathbb{C})}<n.\) The second result gives the explicit conditions for when cpl \(A_{k,d}^{\text{SL} _{2}(\mathbb{C}) }<15\). binary forms Meulien, M.: Sur la complication des algèbres d'invariants combinants. J.~algebra 284, No. 1, 284-295 (2005) Geometric invariant theory, Group actions on varieties or schemes (quotients), Actions of groups on commutative rings; invariant theory On the complication of algebras of combinant invariants | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In the classical \(1\)-dimensional case there is a one-to-one correspondence between integral projective curves over a field \(k\) with a torsion free sheaf obeying some geometric properties and Schur pairs, i.e. pairs of \(k\)-subspaces \((W,A)\) of \(V=k((z))\) satisfying a Fredholm condition with respect to the subspace \(V_+=k[[z]]\) such that \(A\) is a \(k\)-subalgebra of \(V\) and \(A\cdot W\subset W\). This is the Krichever correspondence, and in this article the authors obtain a generalization of the Krichever map for algebraic surfaces.
Parshin and Osipov established the Krichever correspondence in higher dimensions. In the \(2\)-dimensional case it starts with a flag \((X\supset V\supset p)\), a vector bundle \(\mathcal F\) of rank \(r\) on \(X\), a formal trivialization \(e_p\) of \(\mathcal F\) at \(p\), and formal local parameters \(u,t\) at \(p\). By these data this correspondence associates the \(k\)-subalgebra \(A\) of \(V=k((u))((t))\) and \(k\)-subspace \(W\) of \(V^{\oplus r}\) with Fredholm condition for all \(i\).
Contrary to the \(1\)-dimensional case not all such pairs of subspaces comes from geometric data. The authors solve this by introducing another type of geometric object called ribbons. The Krichever map is then decomposed into maps
\[
\left\{\begin{matrix}\text{geometric data}\\(X,C,p,\mathcal F,e_p,u,t)\end{matrix}\right\}\subset\left\{\begin{matrix}\text{geometric data}\\ \text{on ribbons}\end{matrix}\right\}\mapsto\left\{\begin{matrix}\text{pairs of subspaces \((W,A))\)}\\ \text{with Fredholm condition}\end{matrix}\right\}.
\]
Ribbons are ringed spaces which are more general than the notion of formal schemes, having some extra features. The authors give a thorough definition of the category of ribbons and proves the necessary properties of these geometrical objects. They also studies sheaves on ribbons and their cohomology called ind-pro-quasicoherent sheaves on ribbons, and they study their coherence property.
The Picard group of a ribbon is studied and its properties are investigated, with interesting results helping to prove the above correspondece. Also a lot of examples is given all the way.
The article is very well written, mostly self contained and with explicit constructions and examples. Krichever correspondence; Fredholm condition; ribbon; formal ribbons; ML-condition; Mittag-Leffler; function of order; Schur pair Kurke, H., Osipov, D., Zheglov, A.: Formal punctured ribbons and two-dimensional local fields. Journal für die reine und angewandte Mathematik (Crelles J.) \textbf{629}, 133-170 (2009) Schemes and morphisms, Generalizations (algebraic spaces, stacks), Structure of families (Picard-Lefschetz, monodromy, etc.) Formal punctured ribbons and two-dimensional local fields | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In the setting of a variety \(X\) admitting a tilting bundle \(T\) we consider the problem of constructing \(X\) as a quiver GIT quotient of the algebra \(A:=\operatorname{End}_X(T)^{\mathrm{op}}\). We prove that if the tilting equivalence restricts to a bijection between the skyscraper sheaves of \(X\) and the closed points of a quiver representation moduli functor for \(A=\operatorname{End}_X(T)^{\mathrm{op}}\) then \(X\) is indeed a fine moduli space for this moduli functor, and we prove this result without any assumptions on the singularities of \(X\). As an application we consider varieties which are projective over an affine base such that the fibres are of dimension 1, and the derived pushforward of the structure sheaf on \(X\) is the structure sheaf on the base. In this situation there is a particular tilting bundle on \(X\) constructed by Van den Bergh, and our result allows us to reconstruct \(X\) as a quiver GIT quotient for an easy to describe stability condition and dimension vector. This result applies to flips and flops in the minimal model program, and in the situation of flops shows that both a variety and its flop appear as moduli spaces for algebras produced from different tilting bundles on the variety. We also give an application to rational surface singularities, showing that their minimal resolutions can always be constructed as quiver GIT quotients for specific dimension vectors and stability conditions. This gives a construction of minimal resolutions as moduli spaces for all rational surface singularities, generalising the G-Hilbert scheme moduli space construction which exists only for quotient singularities. Karmazyn, J., Quiver GIT for varieties with tilting bundles, Manuscripta Math., 154, 1-2, 91-128, (2017) Geometric invariant theory, Representations of quivers and partially ordered sets, Fine and coarse moduli spaces, Rings arising from noncommutative algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Quiver GIT for varieties with tilting bundles | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0723.00028.]
Let \(G\) be either the trivial group or \(\mathbb{Z}/2\mathbb{Z}\). A conic \(G\)- Lagrangian germ is an equivalence class of \(\mathbb{C}^*\)- and \(G\)- equivariant maps of conic \(G\)-manifolds (complex analytic manifolds with commuting \(\mathbb{C}^*\)- and \(G\)-actions) with certain additional properties. The authors classify infinitesimally stable conic \(G\)- Lagrangian germs by use of Legendrean generating families. They apply their results to the Gauss map \(\gamma:\Theta\to\mathbb{P}_ 2^*\) of the theta divisor \(\Theta\) of a smooth complex algebraic curve \(C\) of genus 3. \(\gamma\) lifts to the homogeneous Gauss map \(\Gamma:\mathbb{C}_ e\to T_ 0^*J\) where \(\mathbb{C}_ e\subset T^*J\) is the conormal bundle of \(\Theta\) and \(J\) is the Jacobian variety of \(C\). \(\Gamma\) turns out to be a conic Lagrangian map with \(\mathbb{Z}/2\mathbb{Z}\)-symmetry induced by the \((- 1)\)-involution on \(J\). Main theorem:
If \(C\) is nonhyperelliptic, the homogeneous Gauss map \(\Gamma\) is locally infinitesimally stable if and only if the canonical model of \(C\) in \(\mathbb{P}_ 2\) admits no higher flexes. If \(C\) is hyperelliptic, then \(\Gamma\) is locally infinitesimally stable. infinitesimally stable Gauss map; Lagrangian germ; theta divisor; complex algebraic curve Malcolm R. Adams, Clint McCrory, Theodore Shifrin, and Robert Varley, Symmetric Lagrangian singularities and Gauss maps of theta divisors, Singularity theory and its applications, Part I (Coventry, 1988/1989) Lecture Notes in Math., vol. 1462, Springer, Berlin, 1991, pp. 1 -- 26. Theta functions and abelian varieties, Riemann surfaces; Weierstrass points; gap sequences, Jacobians, Prym varieties, Theta functions and curves; Schottky problem Symmetric Lagrangian singularities and Gauss maps of theta divisors | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 focuses on pairings on the Hochschild cohomology ring of smooth complex projective varieties. There are mainly two such pairings in the literature, namely:
-- The Shklyarov pairing, introduced in the DG-framework in the preprint [\textit{D. Shklyarov}, ``Hirzebruch-Riemann-Roch theorems for DG-algebras'', \url{arXiv:0710.1937}].
-- The Mukai pairing, defined by \textit{A. Căldăraru} and \textit{S. Willerton} in [New York J. Math. 16, 61--98 (2010; Zbl 1214.14013)] via Serre duality.
Results of \textit{N. Markarian} [J. Lond. Math. Soc., II. Ser. 79, No. 1, 129--143 (2009; Zbl 1167.14005)] and the author [New York J. Math. 14, 643--717 (2008; Zbl 1158.19002)] imply that the Mukai pairing on a variety \(X\) is given (up to sign) via the Hochschild-Kostant-Rosenberg isomorphism by the formula \(<a, b>=\int_X a \wedge b \wedge \mathrm{Td} (X)\). Later on, it has been proved by the author [Mosc. Math. J. 10, No. 3, 629--645 (2010; Zbl 1208.14013)] that the two aforementioned pairings were the same up to signs.
The aim of the current paper is to obtain directly the expression of the Shklyarov pairing. The method relies on the theory of deformation quantization as developed in [\textit{M. Kashiwara} and \textit{P. Schapira}, Deformation quantization modules. Astérisque 345. Paris: Société Mathématique de France (2012; Zbl 1260.32001)] as well as on the index theorem of \textit{P. Bressler, R. Nest} and \textit{B. Tsygan} [Adv. Math. 167, No. 1, 1--25 (2002; Zbl 1021.53064); ibid. 167, No. 1, 26--73 (2002; Zbl 1021.53065)]. Using this, the proof reduces to prove that the Euler class of the structural sheaf \(\mathcal{O}_X\) is the Todd class of \(X\). This fact has been conjectured by Kashiwara in 1991 and proved by the reviewer in [J. Differ. Geom. 90, No. 2, 267--275 (2012; Zbl 1247.32013)]. A completely different proof is presented here. Hochschild homology; Mukai pairing; Riemann-Roch theorem; deformation quantization Riemann-Roch theorems, Chern characters, Riemann-Roch theorems, Deformation quantization, star products A variant of the Mukai pairing via deformation quantization | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be an almost simple simply-connected algebraic group over \(\mathbb{C}\) with the Lie algebra \({\mathfrak g}\). Let \({\mathfrak h}\) be a Cartan subalgebra of \({\mathfrak g}\). We assume \(G\) is of type ADE, as there arise technical issues for type BCFG. (We will remark on, them at relevant places. See footnotes 5 and 12.) At some points, particularly in this introduction, we want to include the case \(G= GL(r)\). We will not make a clear distinction between the case \(G= SL(r)\) and \(GL(r)\) in the main text.
Let \(G_c\) denote a maximal compact subgroup of \(G\). Our main players are
\[
{\mathcal U}^d_G= \text{the Uhlenbeck partial compactification}
\]
of the moduli spaces of framed \(G_c\)-instantons on \(S^4\) with instanton number \(d\). A framing means a trivialization of the fiber of the \(G_c\)-bundle at \(\infty\in S^4\). Framed instantons on \(S^4\) are also called instantons on \(\mathbb{R}^4\), as they extend across \(\infty\) if their curvature is in \(L^2(\mathbb{R}^4)\). We follow this convention. The Uhlenbeck compactifications were first considered in a differential geometric context by Uhlenbeck, Donaldson and others, for more general 4-manifolds and usually without framing. Since we are considering framed instantons, we only get partial compactifications.
We consider Uhlenbeck partial compactifications of instanton moduli spaces on \(\mathbb{R}^4\) as objects in geometric representation theory. We study their intersection cohomology groups and perverse sheaves in view of the representation theory of the affine Lie algebra of \({\mathfrak g}\) or the closely related \({\mathcal W}\)-algebra. We will be concerned only with a very special 4-manifold, i.e., \(\mathbb{R}^4\) (or \(\mathbb{C}^2\) as we will use an algebro-geometric framework). On the other hand, we will study instantons for any group \(G\), while \(G_c= SU(2)\) is usually enough for topological applications.
We will drop `Uhlenbeck partial compactification' hereafter unless it is really necessary, and simply say instanton moduli spaces or moduli spaces of framed instantons, as we will keep `\(U\)' in the notation.
We will study equivariant intersection cohomology groups of instanton moduli spaces
\[
IH^*_{T\times\mathbb{C}^x\times \mathbb{C}^x}({\mathcal U}^d_G),
\]
where \(T\) is a maximal torus of \(G\) acting by change of framing, and \(\mathbb{C}^x\times\mathbb{C}^x\) is a maximal torus of \(GL(2)\) acting on \(\mathbb{R}^4= \mathbb{C}^2\). The \(T\)-action has been studied in the: original context: it is important to understand singularities of instanton moduli spaces around reducible instantons. The \(\mathbb{C}^x\times \mathbb{C}^x\)-action is specific for \(\mathbb{R}^4\); nevertheless, it makes the tangent bundle of \(\mathbb{R}^4\) nontrivial, and yields a meaningful counterpart of Donaldson invariants, as Nekrasov partition functions. (See below.)
More specifically, we will explain the author's joint work with \textit{A. Braverman} et al. [Instanton moduli spaces and \({\mathcal W}\)-algebras. Paris: Société Mathématique de France (SMF) (2016; Zbl 1435.14001)] with an emphasis on its geometric part in these lectures. The stable envelope introduced by \textit{D. Maulik} and \textit{A. Okounkov} [``Quantum groups and quantum cohomology'', Preprint, (2012; \url{arXiv:1211.1287})] and its reformulation in [\textit{H. Nakajima}, in: Symmetries, integrable systems and representations. Proceedings of the conference on infinite analysis: frontier of integrability, Tokyo, Japan, July 25--29, 2011 and the conference on symmetries, integrable systems and representations, Lyon, France, December 13--16, 2011. London: Springer. 403--428 (2013; Zbl 1309.17010)] via Braden's hyperbolic restriction functors are key technical tools. They also in other situations in geometric representation theory. Therefore they will be explained in a general framework. In a sense, a purpose of lectures is to explain these important techniques and their applications. Park City Mathematics Institute Nakajima, Hiraku, Lectures on perverse sheaves on instanton moduli spaces. Geometry of moduli spaces and representation theory, IAS/Park City Math. Ser. 24, 381-436, (2017), Amer. Math. Soc., Providence, RI Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Vertex operators; vertex operator algebras and related structures, Research exposition (monographs, survey articles) pertaining to algebraic geometry Lectures on perverse sheaves on instanton moduli 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 The aim of this paper is to investigate components and singularities of Quot schemes and varieties of commuting matrices. The authors classify its components for any number of matrices of size at most \(7\). They prove that starting from quadruples of \(8\times 8\) matrices, this scheme has generically non reduced components, while up to degree \(7\) it is generically reduced.
Their approach is to recast the problem as deformations of modules and generalize an array of methods: apolarity, duality and Bialynicki-Birula decompositions to this setup. The authors include a thorough review of their methods to make the paper self-contained and accessible to both algebraic and linear-algebraic communities. The results obtained in this paper give the corresponding statements for the Quot schemes of points, in particular the authors classify the components of Quot\(_d(\mathcal{O}_{\mathbb{A}^n}^{\oplus r})\) for \(d\leq7\) and all \(r,n\).
This paper is organized as follows. The first Section is an introduction to the subject and statement of the results. Section 2 deals with notation and Section 3 with some preliminaries. Section 4 concerns structural results on the variety \(C_n(\mathbf{M}_d)\) of \(n\)-tuples of commuting \(d\times d\) matrices and Quot\(^d_r\). Section 5 is devoted to Bialynicki-Birula decompositions and components of Quot\(^d_r\) and Section 6 to some results specific for degree at most eight. The paper is supported by an appendix concerning a functorial approach to comparison between \(C_n(\mathbf{M}_d)\) and Quot\(^d_r\). Quot schemes; varieties of commuting matrices; components and singularities Parametrization (Chow and Hilbert schemes), Relationships between algebraic curves and integrable systems Components and singularities of Quot schemes and varieties of commuting matrices | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 establishes a correspondence between three sets of objects: {\parindent=0.7cm\begin{itemize}\item[--] \(n\)-dimensional varieties that are 3-rationally connected by cubics; \item[--] \(n\)-dimensional Jordan algebras of rank 3; and \item[--] quadro-quadric Cremona transformations of \(\mathbb{P}^{n-1}\).
\end{itemize}} This correspondence respects the natural equivalence relations in the three sets.
A variety \(X\) is 3-rationally connected by cubics if for all 3 general points in \(X\) there is a cubic curve contained in \(X\) passing through the points.
A Jordan algebra is a commutative complex algebra \(J\) such that for all \(x,y\in J\)
\[
x^2(xy)=x(x^2 y).
\]
The rank of \(J\) is the complex dimension of \(<x>\) for \(x\) in a Zariski open set of \(J\).
A quadro-quadric Cremona transformation of \(\mathbb{P}^{n-1}\) is a birational map
\[
F: \mathbb{P}^{n-1}\dasharrow \mathbb{P}^{n-1}
\]
such that the coordinates of \(F\) and \(F^{-1}\) are homogeneous polynomials of degree 2.
The functions between the three sets are given explicitly. Previous works of the authors on the subject are [\textit{L. Pirio} and \textit{F. Russo} , Ann. Inst. Fourier 64, No. 1, 71--111 (2014; Zbl 1310.14021); Int. J. Math. 24, No. 13, Article ID 1350105, 33 p. (2013; Zbl 1311.14017)]. One of the goals of these works was giving new examples of quadro-quadric Cremona transformations.
In the last section it is proved that the correspondence preserves some special properties of the objects: varieties that are Cartesian products correspond to algebras that are direct products and to elementary quadro-quadric transformations. Also, smooth varieties correspond to semisimple algebras and to semispecial Cremona transformations (that is, transformations whose base locus is smooth). Finally, some applications are given to the study of homaloidal cubic polynomials. Jordan algebra; quadro-quadric Cremona transformation; variety 3-connected by cubics Pirio, L., Russo, F.: The \(XJC\)-correspondence. J. Reine Angew. Math. (2014). 10.1515/crelle-2014-0052 Projective techniques in algebraic geometry, Birational automorphisms, Cremona group and generalizations, Associated manifolds of Jordan algebras, Rationally connected varieties The \textit{XJC}-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 For a simple finite dimensional Lie algebra \(\widehat{g}\) of type ADE, let \(g\) be the corresponding (untwisted) affine Lie algebra and \(U_q(\widehat g)\) its quantum affine algebra. In this paper, the author studies finite dimensional representations of \(U_q(\widehat g)\) using geometry of quiver varieties. His purpose is to solve the following conjecture affirmatively, that is, an equivariant \(K\)-homology group of the quiver variety gives the quantum affine algebra \(U_q(\widehat g)\), and to derive results whose analogues are known for \(H_q\).
In \S 1, the author recalls a new realization of \(U_q(\widehat g)\), called Drinfeld realization and introduces the quantum loop algebra \(U_q(Lg)\) as a subquotient of \(U_q(\widehat g)\), which will be studied rather than \(U_q(\widehat g)\). The basic results are recalled on finite dimensional representations of \(U_\varepsilon(Lg)\). And, several useful concepts are introduced.
In \S 2, the author introduces two types of quiver varieties \({\mathcal M}(w)\) and \({\mathcal M}_0(\infty, w)\) as analogues of \(T^*{\mathcal B}\) and the nilpotent cone \(\mathcal N\) respectively. Their elementary properties are given.
In \S 3--\S 8, the author prepares some results on quiver varieties and \(K\)-theory which will be used in later sections.
In \S 9--\S 11, the author considers an analogue of the Steinberg variety
\[
Z(w) = {\mathcal M}(w)\times _{{\mathcal M}_0(\infty,w)}{\mathcal M}(w)
\]
and its equivariant \(K\)-homology \(K^{G_w\times \mathbb{C}^*}(Z(w))\). An algebra homomorphism is constructed from \(U_q(Lg)\) to \(K^{G_w\times \mathbb{C}^*}(Z(W)) \otimes_{\mathbb{Z}[q,q^{-1}]}\mathbb{Q}(q)\).
In \S 12, the author shows that the above homomorphism induces a homomorphism from \(U^\mathbb{Z}_q(Lg)\) to \(K^{G_w\times \mathbb{C}^*}(Z(w))/\text{torsion}\).
In \S 13, the author introduces a standard module \(M_{x,a}\). Thanks to a result in \S 7, it is proved to be isomorphic to \(H_*({\mathcal M}(w)^A_x,\mathbb{C})\) via the Chern character homomorphism. Also, it is shown that \(M_{x,a}\) is a finite dimensional \(l\)-highest weight module. It is conjectured that \(M_{x,a}\) is a tensor product of \(l\)-fundamental representations in some order, which is proved when the parameter is generic in \S 14.1.
In \S 14, it is verified that the standard modules \(M_{x,a}\) and \(M_{y,a}\) are isomorphic if and only if \(x\) and \(y\) are contained in the same stratum. Furthermore, the author shows that the index set \(\{\rho\}\) of the stratum coincides with the set \({\mathcal P} =\{P\}\) of \(l\)-dominant \(l\)-weights of \(M_{0,a}\), the standard module corresponding to the central fiber \(\pi^{-1}(0)\). And, the multiplicity formula \([M(P) : L(Q)] =\dim H^*(i^!_x IC({\mathcal M}^{\text{reg}}_0(\rho_Q)))\) is proved. The result here is simpler than the case of the affine Hecke algebra: nonconstant local systems never appear.
Let \(\text{Res }M(P)\) be the restriction of \(M(P)\) to a \(U_\varepsilon(g)\)-module. In \S\ 15, the author shows the multiplicity formula \([\text{Res }M(P) : L(w - v)] \dim H^*(i_x^! IC({\mathcal M}^{\text{reg}}_0(v, w)))\). This result is compatible with the conjecture that \(M(P)\) is a tensor product of \(l\)-fundamental representations since the restiction of an \(l\)-fundamental representation is simple for type \(A\), and Kostka polynomials give tensor product decompositions.
Two examples are given where \({\mathcal M}^{\text{reg}}_0(v, w)\) can be described explicitly.
As mentioned in the Introduction of this paper, \(U_q(\widehat{g})\) has another realization, called the Drinfeld new realization, which can be applied to any symmetrizable Kac-Moody algebra \(g\), not necessarily a finite dimensional one. This generalization also fits the result in this paper, since quiver varieties can be defined for arbitrary finite graphs. If finite dimensional representations are replaced by \(l\)-integrable representations, parts of the result in this paper can be generalized to a Kac-Moody algebra \(g\), at least when it is symmetric.
If equivariant \(K\)-homology is replaced by equivariant homology, one should get the Yangian \(Y(g)\) instead of \(U_q(\widehat{g})\). The conjecture is motivated again by the analogy of quiver varieties with \(T^*\mathcal B\). As an application, the affirmative solution of the conjecture implies that the representation theory of \(U_q(\widehat g)\) and that of the Yangian are the same. quantum affine algebra; quiver variety; equivariant \(K\)-theory; finite dimensional representation Nakajima, H., Quiver varieties and finite-dimensional representations of quantum affine algebras, \textit{J. Amer. Math. Soc.}, 14, 1, 145-238, (2001) Quantum groups (quantized enveloping algebras) and related deformations, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Group actions on varieties or schemes (quotients), Representations of quivers and partially ordered sets, Connections of basic hypergeometric functions with quantum groups, Chevalley groups, \(p\)-adic groups, Hecke algebras, and related topics Quiver varieties and finite dimensional representations of quantum affine 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 semisimple algebraic group defined over \(\mathbb {Q}_p\), and let \(\Gamma \) be a compact open subgroup of \(G(\mathbb {Q}_p)\). We relate the asymptotic representation theory of \(\Gamma \) and the singularities of the moduli space of \(G\)-local systems on a smooth projective curve, proving new theorems about both:
\begin{itemize}
\item[(1)] We prove that there is a constant \(C\), independent of \(G\), such that the number of \(n\)-dimensional representations of \(\Gamma \) grows slower than \(n^{C}\), confirming a conjecture of \textit{M. Larsen} and \textit{A. Lubotzky} [J. Eur. Math. Soc. (JEMS) 10, No. 2, 351--390 (2008; Zbl 1142.22006)]. In fact, we can take \(C=3\cdot {{\mathrm{dim}}}(E_8)+1=745\). We also prove the same bounds for groups over local fields of large enough characteristic.
\item[(2)]We prove that the coarse moduli space of \(G\)-local systems on a smooth projective curve of genus at least \(\lceil C/2\rceil +1=374\) has rational singularities.
\end{itemize}
For the proof, we study the analytic properties of push forwards of smooth measures under algebraic maps. More precisely, we show that such push forwards have continuous density if the algebraic map is flat and all of its fibers have rational singularities. 1 A. Aizenbud and N. Avni, 'Representation growth and rational singularities of the moduli space of local systems', \textit{Invent. Math.}204 (2016) 245-316. MR 3480557. Algebraic moduli problems, moduli of vector bundles, Singularities in algebraic geometry, Asymptotic properties of groups, Linear algebraic groups over local fields and their integers, Deformations of singularities, Symplectic structures of moduli spaces Representation growth and rational singularities of the moduli space of local systems | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The Hilbert scheme and the Quot scheme, both constructed by Grothendieck, are fundamental objects in algebraic geometry: the first one parametrizes the subschemes of a projective space with fixed Hilbert polynomial, the second one the quotients with fixed Hilbert polynomial of a fixed coherent sheaf on a projective space.
Here the author considers an affine variety \(X\), with an action of a reductive group \(G\), and a coherent sheaf \(\mathcal M\) on \(X\), that is supposed to be \(G\)-linearized. First of all he proves the existence of a quasi-projective scheme, the invariant Quot scheme, parametrizing the quotients \(\mathcal L\) of \(\mathcal M\), such that the space of global sections \(H^0(X,\mathcal L)\) is a direct sum of simple \(G\)-modules with fixed finite multiplicities: the datum of these multiplicities is here the analogous of the Hilbert polynomial. The invariant Quot scheme is a natural generalization of the invariant Hilbert scheme recently introduced by \textit{V. Alexeev} and \textit{M. Brion} [J. Algebr. Geom. 14, 83--117 (2005; Zbl 1081.14005)]. The construction relies on the multigraded Quot scheme of Haiman and Sturmfels, corresponding to the case in which \(G\) is a torus [cf. \textit{M. Haiman} and \textit{B. Sturmfels}, J. Algebr. Geom. 13, No. 4, 725--769 (2004; Zbl 1072.14007)].
In the second part of the article, the author focuses on a special example, the cone \(X\) of primitive vectors of a simple \(G\)-module, with a free sheaf \(\mathcal M\) generated by another simple \(G\)-module. He proves that, in this case, the invariant Quot scheme has only one point, and that it is reduced unless \(X\) is the cone of the primitive vectors of a quadratic vector space \(V\) of odd dimension \(2n+1\) and \(G= \text{Spin}(2n+1)\times H\), for a connected reductive group \(H\). The Quot scheme of this example is isomorphic to \(\text{Spec}({\mathbb C}[t]/\langle t^2\rangle)\). Hilbert scheme; Quot scheme; reductive group; primitive vector Jansou, S., Le schéma quot invariant, J. Algebra, 306, 2, 461-493, (2006) Parametrization (Chow and Hilbert schemes), Geometric invariant theory, Group actions on varieties or schemes (quotients) The invariant Quot scheme | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(\mathbb{B}\) be an indefinite quaternion algebra over \(\mathbb{Q}\), and let \({\mathcal O}\) be an Eichler order of \(\mathbb{B}\). If \(Nr\) denotes the reduced norm of \(\mathbb{B}\), then \(\Gamma = \{\gamma \in {\mathcal O} \mid Nr (\gamma) = 1\}\) is a discrete subgroup of \(\mathbb{B}^{(1)}\cong SL_2 (\mathbb{R})\), and the quotient \(\Gamma \backslash {\mathcal H}\) of the upper half plane \({\mathcal H}\) by \(\Gamma\) is the \(\mathbb{C}\)-valued points of the Shimura curve \(S\) attached to \({\mathcal O}\). It is well-known that \(S(\mathbb{C})\) can be interpreted as the moduli space of principal polarized abelian surfaces having quaternion multiplication by \({\mathcal O}\), and the associated correspondence between the points of \(S(\mathbb{C})\) and the abelian surfaces can also be described by a quaternion modular embedding. More precisely, if \({\mathcal H}_2\) denotes the Siegel upper half space of degree two, then there is a holomorphic embedding \(\Phi : {\mathcal H} \to {\mathcal H}_2\), \(z \mapsto \Omega (z)\) that is compatible with the actions of \(\Gamma\) and \(Sp(4, \mathbb{Z})\) through an embedding \(\varphi : \Gamma \hookrightarrow Sp(4, \mathbb{Z})\). In this paper the author describes the maps \(\varphi\) and \(\Phi\) explicitly by constructing a concrete model of \({\mathcal O}\) and using its \(\mathbb{Z}\)-basis. He also determines some arithmetic properties of such embeddings. Humbert surfaces; quaternion algebra; Shimura curve; abelian surfaces; quaternion modular embedding Hashimoto, K. -I.: Explicit form of quaternion modular embeddings. Osaka J. Math. 32, 533-546 (1995) Arithmetic aspects of modular and Shimura varieties, Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms, Modular and Shimura varieties, Algebraic moduli of abelian varieties, classification Explicit form of quaternion modular embeddings | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The purpose of this paper is to present a short elementary proof of a theorem due to \textit{G. Faltings} [J. Algebr. Geom. 2, 507--568 (1993; Zbl 0790.14019)] and \textit{G. Laumon} [Duke Math. J. 57, 647--671 (1988; Zbl 0688.14023)], which says that the global
nilpotent cone is a Lagrangian substack in the cotangent bundle of the moduli space of \(G\)-bundles on a complex compact curve. This result plays a crucial role in the geometric Langlands program [\textit{A.~A. Beilinson} and \textit{V.~G. Drinfeld}, in: Algebraic and geometric methods in mathematical physics. Proc. 1st Ukrainian-French-Romanian summer school, Kaciveli, Ukraine 1993. Math. Phys. Stud. 19, 3--7 (1996; Zbl 0864.14007)] since it insures that the \(\mathcal D\)-modules on the moduli space of \(G\)-bundles whose characteristic variety is contained in the global nilpotent cone are automatically holonomic and, in particular, have finite length. Ginzburg V., The global nilpotent variety is Lagrangian, Duke Math. J., 2001, 109(3), 511--519 Algebraic moduli problems, moduli of vector bundles, Lagrangian submanifolds; Maslov index The global nilpotent variety is Lagrangian. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 setting in this article fixes a simple complex Lie algebra \({\mathfrak g}\), a Cartan subalgebra \({\mathfrak h}\), a root system \(Q\) and a Killing form, normalized such that the longest root is of length 2. The set of dominant integral weights of level \(\ell\) is denoted \(P_{\ell}({\mathfrak g})\). For an \(n\)-tuple \(\vec{\lambda}\) of weights, the corresponding bundle of conformal blocks on the moduli stack \(\overline{\mathrm M}_{g,n}\) of genus \(g\) curves with \(n\) marked points is denoted \({\mathbb V}_{\vec{\lambda}}({\mathfrak g},\ell)\).
Rank-level duality is a duality in the special case \(g=0\) between certain conformal blocks of \({\mathfrak s}{\mathfrak l}(r)\) at level \(s\) and certain conformal blocks of \({\mathfrak s}{\mathfrak l}(s)\) at level \(r\). There exist similar results for the symplectic and odd orthogonal Lie algebras.
The goal of the article under review is to relate explicitely the conformal block divisors, i.e. the Chern classes \(c_1({\mathbb V}_{\vec{\lambda}}({\mathfrak g},\ell)) \), on \(\overline{\mathrm M}_{g,n}\) using the rank-level duality isomorphisms. Under some technical conditions on \(\vec{\Lambda}\in P_1({\mathfrak g})^n\), \(\vec{\lambda}\in P_{\ell_1}({\mathfrak g}_1)^n\) and \(\vec{\mu}\in P_{\ell_2}({\mathfrak g}_2)^n\) which are known to be satisfied in the case of conformal embeddings \({\mathfrak s}{\mathfrak l}(r)\oplus{\mathfrak s}{\mathfrak l}(s)\to {\mathfrak s}{\mathfrak l}(rs)\) and \({\mathfrak s}{\mathfrak p}(2r)\oplus{\mathfrak s}{\mathfrak p}(2s)\to {\mathfrak s}{\mathfrak o}(4rs)\), the main rsult of the article (Theorem 1.2) asserts the following relation among conformal block divisors in the Picard group \(\mathrm{Pic}(\overline{\mathrm M}_{0,n})\):
\[
c_1({\mathbb V}_{\vec{\lambda}}({\mathfrak g}_1,\ell_1))+ c_1({\mathbb V}_{\vec{\mu}}({\mathfrak g}_2,\ell_2)) = \mathrm{rk}{\mathbb V}_{\vec{\lambda}}({\mathfrak g}_1,\ell_1).\left\{ c_1({\mathbb V}_{\vec{\Lambda}}({\mathfrak g},1))+\sum_{j=1}^{n} n_{\lambda_j,\mu_j}^{\Lambda_j}\psi_j\right\}
\]
\[
- \sum_{i=2}^{\;lfloor\frac{n}{2}\rfloor}\epsilon_i\left\{\sum_{A\subset\{1,\ldots,n\},|A|=i} b_{A,A^c}[D_{A,A^c}]\right\},
\]
where \([D_{A,A^c}]\) denotes the class of the boundary divisor corresponding to the partition \(A\cup A^c=\{1,\ldots,n\}\), \(\epsilon_i=\frac{1}{2}\) if \(i=n/2\) and \(\epsilon_i=1\) otherwise, \(\psi_j\) is the \(j\)th psi class, \(n_{\lambda_j,\mu_j}^{\Lambda_j}\) is an integer related to the conformal embedding (a sum/difference of trace anomalies), and \(b_{A,A^c}\) is another integer (sum of products of ranks of conformal blocks with some \(n_{\lambda,\mu}^{\Lambda}\) as coefficients).
The author proposes two proofs of the above formula, one using a geometric approach (vertex algebra techniques) and one using Fakhruddin's Chern class formula. rank-level duality; vertex algebras; conformal blocks; Picard group of the moduli stack of stable curves; psi classes; conformal embedding S. Mukhopadhyay, Rank-level duality and conformal block divisors, preprint (2013), . Vector bundles on curves and their moduli, Families, moduli of curves (algebraic), Relationships between algebraic curves and physics, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Vertex operators; vertex operator algebras and related structures, Two-dimensional field theories, conformal field theories, etc. in quantum mechanics Rank-level duality and conformal block divisors | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(K\) be a field, \(G\) a finite group and \(V\) a faithful representation of \(G\) over \(K\). Then there is a natural action of \(G\) upon the field of rational functions \(K(V)\). The rationality problem (known as Noether's problem when \(G\) acts on \(V\) by permutations) asks whether the field of \(G\)-invariant functions \(K(V)^G\) is rational (purely transcendental) over \(K\). The Bogomolov multiplier \(B_0(G)\) of a finite group \(G\) is defined as the subgroup of the Schur multiplier consisting of the cohomology classes vanishing after restriction to all abelian subgroups of \(G\). The triviality of the Bogomolov multiplier is an obstruction to Noether's problem. The author shows that if \(G\) is a central product of \(G_1\) and \(G_2\), regarding \(K_i \leq Z(G_i)\) , \(i = 1, 2\), and \(\theta:G_1\rightarrow G_2\) is a group homomorphism such that its restriction \(\theta_{|K_1}:K_1\rightarrow K_2\) is an isomorphism, then the triviality of \(B_0(G_1/K_1)\), \(B_0(G1)\) and \(B_0(G_2)\) implies the triviality of \(B_0(G)\). He gives a positive answer to Noether's problem for all 2-generator \(p\)-groups of nilpotency class 2, and for one series of 4-generator \(p\)-groups of nilpotency class 2 (with the usual requirement for the roots of unity). Bogomolov multiplier; Noether's problem; rationality problem; central product of groups; \(p\) -groups of nilpotency class 2 I. M. Michailov, Bogomolov multipliers for some \(p\)-groups of nilpotency class 2, arXiv:1307.0738. Rationality questions in algebraic geometry, Rational and unirational varieties, Actions of groups on commutative rings; invariant theory, Inverse Galois theory Bogomolov multipliers for some \(p\)-groups of nilpotency class 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 Let \(f: (\mathbb{C}^{n+1},0) \to(\mathbb{C},0)\) be a germ of a holomorphic function with an isolated critical point at the origin, \(V_\varepsilon =f^{-1} (\varepsilon) \cap B_\delta\) -- its Milnor fiber \((0<|\varepsilon |\ll \delta\); \(B_\delta\) is the ball of radius \(\delta\) with center at the origin in \(\mathbb{C}^{n+1})\). The matrix of the intersection form on the homology group \(H_n (V_\varepsilon, \mathbb{Z})\) in a basis of a special type (so-called distinguished basis) or the Dynkin diagram (the graphic representation of this matrix) is of interest for study of hypersurface singularities. For calculation of Dynkin diagrams the method of real morsifications is known. It can be applied only to singularities of functions of two variables (or the case \(n=1)\), and expresses a Dynkin diagram in terms of the geometry of a real plane curve with simple self-intersections.
In this article the generalization of the method of real morsifications for one-dimensional isolated complete intersection singularities is explained. The generalization is almost parallel to the hypersurface case except that a reasonable analogue of a distinguished basis is not a basis of \(H_n (V_\varepsilon, \mathbb{Z})\) but a set of generators. Several concrete examples are given in the latter half. Milnor fiber; Dynkin diagram; real morsifications; complete intersection singularities S.M. Gusein-Zade, ''Dynkin diagrams of some complete intersections and real morsifications,''Tr. Mat. Inst. Russ. Akad. Nauk (in press). Local complex singularities, Milnor fibration; relations with knot theory, Singularities of curves, local rings, Complete intersections, Complex surface and hypersurface singularities, Singularities in algebraic geometry Dynkin diagrams of some complete intersections and real morsifications | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 $d\ge 1$ let $X_d$ be the following Gorenstein threefold singularity: \[ X_d:=\text{Spec} (\mathbb{C})[x,y,z,w]/ (x^2 +y^2 +(z+w^d)(z-w^d)). \] Let $Y_d$ be the resolution of singularities obtained by blowing up the curve $0=x=z+w^d$. The exceptional curve $C_d\cong \mathbb P^1$ has normal bundle $\mathcal O(-1)\oplus \mathcal O(-1)$ and width $d$ (in the sense of Reid). \par The paper under review computes the motivic Donaldson-Thomas invariants (in the sense of Kontsevich-Soibelman) defined by means of the derived category of compactly supported coherent sheaves on $Y_d$. They depend on orientation data and take values in the ring $K^{\hat \mu}(\text{Var} / \text{Spec}(\mathbb C))[\mathbb L^{-1/2}]$, where $K^{\hat \mu}(\text{Var} / \text{Spec}(\mathbb C))$ is the Grothendieck ring of $\hat \mu$-equivariant varieties. Consider the pair $(n,m)$ associated to the cohomology class $(n-m)[C_d]+m[\text{pt}]$. Then the motivic DT invariants corresponding to the pair $(n,m)$ are given by \[ \begin{cases} \mathbb L^{-1/2}(1-[\mu_{d+1}]) & \text{ if } (n,m)=(n,n+1) \text{ or } (n+1,n) \\ \mathbb P^1\cdot \mathbb L^{3/2} & \text{ if } (n,m)=(n,n). \end{cases} \] Here, $\mu_{d+1}$ is considered as a $\mu_{d+1}$-equivariant variety in the natural way. So this calculation has produced an example of motivic Donaldson-Thomas invariants with nontrivial monodromy action. Donaldson-Thomas theory; minus two curves; motivic invariants Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) The motivic Donaldson-Thomas invariants of \((-2)\)-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 We give an explicit minimal graded free resolution, in terms of representations of the symmetric group \(S_d\), of a Galois-theoretic configuration of \(d\) points in \(\mathbf{P}^{d-2}\) that was studied by Bhargava in the context of ring parametrizations. When applied to the geometric generic fiber of a simply branched degree \(d\) cover of \(\mathbf{P}^1\) by a relatively canonically embedded curve \(C\), our construction gives a new interpretation for the splitting types of the syzygy bundles appearing in its relative minimal resolution. Concretely, our work implies that all these splitting types consist of scrollar invariants of resolvent covers. This vastly generalizes a prior observation due to Casnati, namely that the first syzygy bundle of a degree 4 cover splits according to the scrollar invariants of its cubic resolvent. Our work also shows that the splitting types of the syzygy bundles, together with the multi-set of scrollar invariants, belong to a much larger class of multi-sets of invariants that can be attached to \(C\to\mathbf{P}^1 \): one for each irreducible representation of \(S_d\), i.e., one for each partition of \(d\). covering of curves; gonality; syzygy; Galois coverings of curves; scrollar invariants. Scrollar invariants, syzygies and representations of the symmetric group | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(\mathfrak g\) be a noncommutative reductive Lie algebra. The commuting variety \(\mathcal C=\mathcal C(\mathfrak g)\subset \mathfrak g\times \mathfrak g\) is defined as the set of pairs of commuting elements: \(\mathcal C=\{(x,y): [x,y]=0\}\). This paper is devoted to the study of the singularities of \(\mathcal C\).
The author proves that the variety \(\mathcal C\) is always singular, and obtains a lower bound for the codimension of the singular locus \(\mathcal C^{\mathrm{sing}}\subset \mathcal C\). Namely, he shows that \(\mathrm{codim}_{\mathcal C}\mathcal C^{\mathrm{sing}}\geq 5-l\), where \(l=l(\mathfrak g)\) is the ``lacety'' of \(\mathfrak g\): i.e., \(l=1\) for \(\mathfrak g\) simply-laced, \(l=2\) for a non-simply laced algebra \(\mathfrak g\) not containing simple ideals of type \(\mathrm G_2\), and 3 otherwise. In particular, \(\mathrm{codim}_{\mathcal C}\mathcal C^{\mathrm{sing}}\) is always greater than or equal to 2, so this may be considered as an evidence in favor of the long-standing conjecture stating the normality of \(\mathcal C\).
To prove this, the author considers the natural action of the adjoint group \(G\) of \(\mathfrak g\) on \(\mathcal C\) and defines the set \(\mathcal C^{\mathrm{irr}}\subset\mathcal C\) of irregular points as the set of points inside \(\mathcal C\) whose \(G\)-stabilizers for this action have the dimension greater than \(\mathrm{rk}\; G\), i.e. the dimension of the stabilizer for a generic point in \(\mathcal C\). Then he proves the inclusion \(\mathcal C^{\mathrm{sing}}\subset\mathcal C^{\mathrm{irr}}\) and estimates the dimension of \(\mathcal C^{\mathrm{irr}}\) by Lie-algebraic methods. Finally, he proves that the variety \(\mathcal C\) is always rational. commuting variety; singular locus; decomposition class; irregular element; semisimple element; nilpotent element \beginbarticle \bauthor\binitsV. L. \bsnmPopov, \batitleIrregular and singular loci of commuting varieties, \bjtitleTransform. Groups \bvolume13 (\byear2008), page 819-\blpage837. \endbarticle \OrigBibText V. L. Popov, Irregular and singular loci of commuting varieties , Transform. Groups, 13 (2008), 819-837. \endOrigBibText \bptokstructpyb \endbibitem Lie algebras of linear algebraic groups, Special varieties, Group actions on varieties or schemes (quotients), Group actions on affine varieties, Geometric invariant theory Irregular and singular loci of commuting varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The authors review the geometric set-up for the computation of superpotential, and discuss the relation to charge vectors. Then, they derive the extended charge vectors and prove the quantum McKay correspondence of disc invariants for branes intersecting effective outer legs. In addition, using Ooguri-Vafa invariants the authors discuss some enumerative meaning for integrality of open-closed mirror maps. Finally, some examples are presented. gauged linear sigma model; disc invariants; Gromov-Witten invariants; mirror symmetry; open string; charge vectors; Picard-Fuchs system Ke, H.-Z., Zhou, J.: Gauged linear sigma model for disc invariants. \textit{Lett. Math. Phys.}, accepted Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Mirror symmetry (algebro-geometric aspects) Gauged linear sigma model for disc invariants | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The article investigates a particular combinatorial structure known as an alternating strand diagram or Postnikov diagram. This is roughly a collection of strands defined on an oriented disc, each with a specified start and end point on the boundary, finitely many intersections with other strands, no self-intersections, and specific laws governing the orientation of intersections. Postnikov diagrams are useful for encoding geometric information, particularly with regard to the Grassmannian, and the main result of the paper concerns the associated dimer algebra of a connected Postnikov diagram.
In short, the main theorem of the article, Theorem 1, is a categorification result, whose underlying algebra is the cluster algebra \(\mathcal{A}_D\) of the associate ice quiver of a Postikov diagram \(D\). The theorem states that the category of Gorenstein-projective modules \(GP(B_D)\) over the boundary \(B_D\) of the dimer algebra of \(D\) can be realised as an additive categorification of this cluster algebra. Since this cluster algebra is closely related to the homogeneous coordinate ring of a positroid variety in the Grassmannian, this result is certainly useful from a geometric point of view. The key step in the argument is the second main result, Theorem 2, which states that the dimer algebra of \(D\) satisfies the appropriate Calabi-Yau property.
The paper begins with a rough outline of the main results and their motivation, focusing heavily on positroid varieties, the current state of their research in the community, and how the results of the article advance this research. After introducing some preliminary material in section 2, including of course the formal definition and basic properties of a Postnikov diagram, its associated ice quiver, the dimer algebra and the cluster algebra, the article goes on in section 3 to explore the Calabi-Yau property. This section focuses on the dimer algebra of a Postnikov diagram, and after describing some algebraic and combinatorial properties of this algebra, it concludes with a complete proof of Theorem 2 (here relabelled Theorem 3.7).
In section 4, the article explores some category theory, and the main theorem (Theorem 4.3) concerns a general algebra satisfying the appropriate Calabi-Yau property, and roughly speaking it states that the associated category of Gorenstein-projective modules is triangulated and satisfies many of the properties required of the categorification of a cluster algebra, and that it contains a specified cluster tilting object. Using Theorem 2, this result of course applies to the dimer algebra.
After proving in section 5 that the quiver associated to the cluster tilting objects in Theorem 4.3 contain no loops or cycles, the article goes on to use this fact to reduce to the situation previously explored by \textit{C. Fu} and \textit{B. Keller} [Trans. Am. Math. Soc. 362, No. 2, 859--895 (2010; Zbl 1201.18007)], and consequently prove Theorem 1 in section 6 (relabelled as Theorem 6.11). Finally, the paper concludes in section 7 with a discussion of the Grassmannian cluster category, and explores how it is closely related to the categorification explored throughout.
Overall, this paper should be of great interest to anyone working with quivers, cluster algebras and interested in their geometric and combinatorial properties. It follows on from previous work of many different authors, none of whom achieved results in as great a generality of this, so it should create a stir in the community. quivers; cluster-algebras; path-algebras; categorification; Grassmannians; triangulated-categories Representations of quivers and partially ordered sets, Cluster algebras, Grassmannians, Schubert varieties, flag manifolds, Abelian categories, Grothendieck categories, Derived categories, triangulated categories Calabi-Yau properties of Postnikov 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 After having fixed two positive integers \(g\geq 3\) and \(d\), one can consider the functor of isomorphism classes of smooth, proper and connected complex curves of genus \(g\), admitting a finite morphism of degree \(d\) to \(\mathbb{P}_{\mathbb{C}}^1\), whose branch divisor is supported at \(2g+2d-2\) distinct points. This moduli problem is known to have a coarse moduli space, denoted by \(\mathcal{H}_{d,g}\), which is a normal, \(\mathbb{Q}\)-factorial and irreducible quasi-projective complex variety of dimension \(2g+2d-5\). The Picard rank conjecture predicts that \(\mathrm{Pic}(\mathcal{H}_{d,g})\otimes \mathbb{Q}=0\). One consequence of this conjecture is the expectation that a certain partial compactification \(\widetilde{\mathcal{H}_{d,g}}\) of \(\mathcal{H}_{d,g}\), obtained by allowing nodal, irreducible curves and non simply branches, should have rational Picard group generated by boundary classes. It is known that the boundary classes can be expressed in terms of ``tautological classes''. In particular the expectation is that \(\mathrm{Pic}(\widetilde{\mathcal{H}_{d,g}})\otimes \mathbb{Q}\) is generated by tautological classes. The validity of the conjecture was known before the paper under review for \(d=2,3\) and for large \(d\), namely \(d>2g-2\). In the paper under review the authors prove this conjecture under the assumption that \(3\leq d\leq 5\) (Theorem A). The strategy of the proof consists in first showing that for \(d\geq 3\) (resp. \(d\geq 4\)) there are at least two (resp. at least three) divisorial components supported on \(\widetilde{\mathcal{H}_{d,g}}\setminus\mathcal{H}_{d,g}\) whose classes are linearly independent in \(\mathrm{Pic}( \widetilde{\mathcal{H}_{d,g}})\otimes \mathbb{Q}\) (Proposition 2.15). The most delicate part is then to show that \(\mathrm{rk}(\mathrm{Pic}( \widetilde{\mathcal{H}_{d,g}})\otimes \mathbb{Q})\leq 2\) (resp. \(\leq 3\)) for \(d=3\) (resp. for \(d=4,5\)). In order to produce these upper bounds the authors find a suitable open \(U\subset \widetilde{\mathcal{H}_{d,g}}\) which can be expressed as successive quotients of an open in a projective space by the action of explicit linear algebraic groups and whose number of divisorial components in \(\widetilde{\mathcal{H}_{d,g}}\setminus U\) can be explicitly computed. The authors associate to each smooth curve \(C\) of genus \(g\), equipped with a degree \(d\) morphism \(C\rightarrow \mathbb{P}_{\mathbb{C}}^1\) an embedding of \(C\) into the projectification of the associated Tschirnhausen bundle over \(\mathbb{P}_{\mathbb{C}}^1\). A resolution of the structure sheaf of the curve via this embedding is computed by using theorem 2.1 in [\textit{G. Casnati} and \textit{T. Ekedahl}, J. Algebr. Geom. 5, No. 3, 439--460 (1996; Zbl 0866.14009)]. The open \(U\subset \widetilde{\mathcal{H}_{d,g}}\) is obtained via a suitable mixing (depending on \(d\)) of the loci of curves where the associated Tschirnausen bundle and the first bundle in the Casnati-Ekedahl resolution are most generic. The loci are compared with certain Severi varieties as considered in [\textit{A. Ohbuchi}, J. Math., Tokushima Univ. 31, 7--10 (1997; Zbl 0938.14011)]. These Severi varieties are defined as follows. Fix \(m\) a positive integer, consider the Hirzebruch surface \(\mathbb{F}_m\) and \(\tau\subset \mathbb{F}_m\) a section with self-intersection equal to \(m\). Define \(\mathcal{V}_g(\mathbb{F}_m,d\tau)\) as the closure in the linear series \(|d\tau|\) of the locus \(\mathcal{U}_g(\mathbb{F}_m,d\tau)\) parametrizing irreducible, nodal curves of genus \(g\) in \(|d\tau|\). Ohbuchi gives a way to produce from a smooth curve of genus \(g\) equipped with a degree \(d\) morphism to \(\mathbb{P}_{\mathbb{C}}^1\) and a positive integer \(m\) a point in the Severi variety \(\mathcal{V}_g(\mathbb{F}_m,d\tau)\).
\noindent As consequence of this association the authors can show that for any \(d\) and \(m\) larger than an explicit number (depending on \(d\) and \(g\)) the Picard rank conjecture is equivalent to \(\mathrm{Pic}(\mathcal{U}_g(\mathbb{F}_m,d\tau))\otimes\mathbb{Q}=0\) (Theorem B). Hurwitz space; Picard group A. Deopurkar and A. Patel, The Picard rank conjecture for the Hurwitz spaces of degree up to five. Available at http://arxiv.org/pdf/1402.1439v2, 2014. Families, moduli of curves (algebraic), Picard groups The Picard rank conjecture for the Hurwitz spaces of degree up to five | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 was published twice, one times in the book Zbl 0527.00017.]
This paper is a sequel to the author's paper ''Fifteen characterizations of rational double points and simple critical points'' [Enseign. Math. 25, 132-163 (1979; Zbl 0418.14020)]. The characterizations of that paper are for complex varieties and complex functions, and involve the Dynkin diagrams \(A_ k\), \(D_ k\) and \(E_ k\). It turns out that the missing Dynkin diagrams \(B_ k\), \(C_ k\), and \(F_ 4\) (but not \(G_ 2)\) correspond to real singularities and real functions, and that a smaller number of similar characterizations are true for these as well. The main theorem of this paper contains four such characterizations: Let \(f:({\mathbb{R}}^ 3,0)\rightsquigarrow(\mathbb{R},0)\) be the germ at the origin of a real analytic function. Then the following are equivalent: (1) The germ f is right-left equivalent to one of the germs given in a certain list. (2) The germ f is simple (in the sense of Arnold). (3) The complexified variety \(f^{-1}(0)\) has a rational singularity at the origin. (4) A resolution of the real variety \(f^{-1}(0)\) is given in a certain list. The proof of the theorem proceeds by direct computation, or by referring to the corresponding theorem in the complex case. germ of a real analytic function; singularities of real varieties; rational double points; simple critical points; Dynkin diagrams; real singularities A. Durfee, 14 characterizations of rational double points (to appear). Singularities in algebraic geometry, Real algebraic and real-analytic geometry, Germs of analytic sets, local parametrization, Singularities of differentiable mappings in differential topology, Local complex singularities Four characterizations of real rational double points | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper under review deals with two related types of associative algebras, the deformed preprojective algebra \(\Pi^\lambda\) and the Lusztig algebra \(L^\lambda\) of a finite (connected) graph/quiver. In characteristic zero the (matrix) Hilbert series of the preprojective algebra \(\Pi^0\) is determined explicitly. This is accomplished by realizing \(\Pi^0\) as preprojective algebra of the modulated graph via the quantum McKay correspondence and by exploiting knowledge of the \(q\)-symmetric algebra. In another result the zeroth Hochschild homology of the deformed preprojective algebra is described in the finite and in the affine cases.
\textit{G.\ Lusztig} described Nakajima's quiver varieties (introduced by \textit{H.\ Nakajima} [in Duke Math. J. 76, No. 2, 365-416 (1994; Zbl 0826.17026); ibid. 91, No. 3, 515-560 (1998; Zbl 0970.17017)]) as certain Grassmannians. For this he defined [in Adv. Math. 136, No. 1, 141-182 (1998; Zbl 0915.17008)] a map \(\vartheta\) from Nakajima's variety (weight \(\lambda\)) to the variety of representations of the algebra \(L^\lambda\), and he proved for quivers of ADE type that this map is finite and a homeomorphism onto its image.
The main result of the present paper is to show that in the ADE case the map \(\vartheta\) is an algebraic isomorphism onto its image. This was known before only in the \(A\) case [see \textit{A.\ Maffei}, Comment. Math. Helv. 80, No. 1, 1-27 (2005; Zbl 1095.16008)]. As an application the (reduced) variety of representations of Lusztig's algebra is shown to be Poisson. preprojective algebras; quiver varieties; tensor categories Malkin A., Ostrik V. and Vybornov M., Quiver varieties and Lusztig's algebra, Adv. Math. 203 (2006), no. 2, 514-536. Representations of quivers and partially ordered sets, Group actions on varieties or schemes (quotients), Grassmannians, Schubert varieties, flag manifolds, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras Quiver varieties and Lusztig's algebra. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be a connected, simple algebraic group of adjoint type over \(\mathbb C\) with corresponding Lie algebra \(\mathfrak g\). The orbit \(\mathcal O\) of a nilpotent element in \(\mathfrak g\) under the adjoint action is called a nilpotent orbit, and its closure \(\overline{\mathcal O}\) is a union of finitely many nilpotent orbits. The closure inclusion is a partial order on the set of nilpotent orbits, and this article considers the generic singularities, the singularities of \(\overline{\mathcal O}\) at points of maximal orbits of its singular locus.
\textit{H. Kraft} and \textit{C. Procesi} [Invent. Math. 62, 503--515 (1981; Zbl 0478.14040)] have determined the generic singularities for the classical types of Lie algebras, \textit{E. Brieskorn} [Actes Congr. internat. Math. 1970, 2, 279-284 (1971; Zbl 0223.22012)] and \textit{P. Slodowy} [``Simple singularities and simple algebraic groups''. Lect. Notes Math. 815 (1980; Zbl 0441.14002)] determined them for the whole nilpotent cones \(\mathcal N\) for \(\mathfrak g\) of any type. The goal of this paper is to determine the generic singularities of \(\overline{\mathcal O}\) when \(\mathfrak g\) is of exceptional type.
A basic result is that the singular locus of \(\overline{\mathcal O}\) coincides with the boundary of \(\mathcal O\) in \(\overline{\mathcal O}\). To study generic singularities of \(\overline{\mathcal O}\), it is sufficient to study each maximal orbit \(\mathcal O^\prime\) in the boundary \(\overline{\mathcal O}\) of \(\mathcal O\). Such an \(\mathcal O^\prime\) is called a \textit{minimal degeneration} of \(\mathcal O\).
The local geometry of \(\overline{\mathcal O}\) at \(e\in\mathcal O^\prime\) is determined by the intersection of \(\overline{\mathcal O}\) with a transverse slice to \(\mathcal O^\prime\) at \(e\) in \(\mathfrak g\). Such slices exists in all cases, given by the affine space \(\mathcal S_e=e+\mathfrak g^f\), called the \textit{Slodowy slice}, where \(e,f\) are the nilpotent parts of an \(\mathfrak{sl}_2 \)-triple, and \(\mathfrak g^f\) is the centralizer of \(f\) in \(\mathfrak g\). The local geometry is thus encoded in \(\mathcal S_{\mathcal O,e}=\overline{\mathcal O}\cap\mathcal S_e\) which is named a \textit{nilpotent Slodowy slice}. If \(\mathcal O^\prime\) is a minimal degeneration of \(\mathcal O\), \(\mathcal S_{\mathcal O,e}\) has an isolated singularity at \(e\), and the generic singularities of \(\overline{\mathcal O}\) can be determined by studying the various \(\mathcal S_{\mathcal O,e}\) as \(\mathcal O^\prime\) runs over all minimal degenerations and \(e\in\mathcal O^\prime\), the isomorphism type of \(\mathcal S_{\mathcal O,e}\) being independent of \(e\).
The main result of the article is a classification of \(\mathcal S_{\mathcal O,e}\) up to algebraic isomorphism for each minimal degeneration \(\mathcal O^\prime\) of \(\mathcal O\) in the exceptional types. In a few cases, the result is able to determine the normalization of \(\mathcal S_{\mathcal O,e}\), and in another few cases, \(\mathcal S_{\mathcal O,e}\) is only determined up to local analytic isomorphism.
The article uses the theory of symplectic varieties. By \textit{Y. Namikawa} [J. Reine Angew. Math. 539, 123--147 (2001; Zbl 0996.53050)], a normal variety is symplectic if and only if its singularities are rational, Gorenstein, and its smooth part carries a holomorphic symplectic form, and it is proved that the normalization of a nilpotent orbit \(\overline{\mathcal O}\) is a symplectic variety. Because the normalization \(\overline{\mathcal O}\) has rational Gorenstein singularities, so has the normalization \(\tilde{\mathcal S}_{\mathcal O,e}\), and it is a symplectic variety. A singularity of a symplectic variety is what is called a \textit{symplectic singularity}, and the authors claim that a better understanding of those is important for studying the conjecture that a Fano contact manifold is homogeneous. Thus it is important to find examples of symplectic singularities, and the study of the isolated symplectic singularity \(\tilde{\mathcal S}_{\mathcal O,e}\) contributes to this.
The results in the article is motivated by representation theory. The geometry of the nilpotent cone \(\mathcal N\) was important in Springer's construction of Weyl group representations and the resulting Springer correspondence. It is proved that modular representation theory of the Weyl group of \(\mathfrak g\) is encoded in the geometry of \(\mathcal N\). Its decomposition matrix is a part of the decomposition matrix for equivariant perverse sheaves on \(\mathcal N\). Also, the authors remark that the reappearance of certain singularities in different nilpotent cones leads to equalities between parts of decomposition matrices. In the \(\text{GL}_n\)-case, the row and column removal rule for nilpotent singularities gives a geometric explanation for a similar rule for decomposition matrices of symmetric groups.
The main results in the article concerns simple surface singularities and their symmetries. A finite subgroup \(\Gamma\subset\text{SL}_2(\mathbb C)\cong\text{Sp}_2\) acts on \(\mathbb C^2\) and the quotient variety is an affine symplectic variety with an isolated singularity at the image of \(0\), known as a simple surface singularity, a double point, a du Val singularity, or a Kleinian singularity. Up to conjugacy in \(\text{SL}_2(\mathbb C)\), such \(\Gamma\) are in one-to-one correspondence with the simply-laced, simple Lie-algebras over \(\mathbb C\). Thus the simple singularities can be denoted as \(A_k\), \(D_k\;(k\geq 4)\), \(E_6\), \(E_7\), \(E_8\), according to the associated simple Lie algebra. The article contains a proof of the fact that in dimension 2, an isolated symplectic singularity is equivalent to a simple surface singularity. Also, the more general case is considered.
An automorphism of \(X=\mathbb C^2/\Gamma\) gives rise to a graph automorphism of the dual graph \(\Delta\) of \(X\). The authors ask when the action of \(\text{Aut}(\Delta)\) on the dual graph comes from an algebraic action on \(X\), and proves for which types of \(X\) this is true.
The article contains the study of the regular nilpotent orbit, starting with the generic singularities of the nilpotent cone.
In proving a conjecture of Grothendieck, Brieskorn and Slodowy [loc. cit.] described the generic singularities of the nilpotent cone \(\mathcal N\) of \(\mathfrak g\). In this particular case, \(\mathcal O\) is the regular nilpotent orbit, and so \(\overline{\mathcal O}\) equals \(\mathcal N\) with only one degeneration at the subregular nilpotent orbit \(\mathcal O^\prime\). Slodowy concludes that when \(e\in\mathcal O^\prime\), the slice \(\mathcal S_{\mathcal O,e}\) is algebraically isomorphic to a simple surface singularity. Also, when the Dynkin diagram of \(\mathfrak g\) is simply-laced, the Lie algebra associated to the simple surface singularity is \(\mathfrak g\). When it is not simply-laced, the singularity \(\mathcal S_{\mathcal O,e}\) belongs to a list given explicitly in the article. The authors explain an intrinsic realization of the symmetry of \(\mathcal S_{\mathcal O,e}\) when \(\mathfrak g\) is not simply-laced.
Kraft and Procesi [loc. cit]. classified the generic singularities of nilpotent orbit closures for all the classical groups, up to smooth equivalence. The \textit{minimal singularities} are those corresponding to the equivalence classes of a particular class of singularities, explicitly defined by their orbit closures, and denoted \(a_k,b_k,c_k,d_k(k\geq 4),g_2,f_4,e_6,e_7,e_8\). The \textit{generic singularities} in the classical types are classified: An irreducible component of a generic singularity is either a simple surface singularity or a minimal singularity, up to smooth equivalence. When a generic singularity is not irreducible, then it is smoothly equivalent to a union of two simple surface singularities of type \(A_{2k-1}\) meeting transversally in the singular point, denoted \(2A_{2k-1}\).
The main goal of the article is to describe the classification of generic singularities in the exceptional Lie algebras. The symmetry of minimal singularities and the generalization to the general case is essential. The exceptional Lie algebras introduce additional singularities, and and the non-normal cases is treated thoroughly. The main theorem classify generic singularities of nilpotent orbit closures in a simple Lie algebra of exceptional type, and graphs at the end of the article list precise results.
The object of the article is very interesting, and its goal is justified. In addition to treating the subject in a very interesting way, the authors introduce new theories, and explain established ones in a good way, making this a good overview of the subject. nilpotent orbits; symplectic singularities; Slodowy slice; Dynkin diagrams; Dynkin classification; transverse slice; generic singularities; degenerate orbits; closure relation; simple singularities,isolated singularities; simple surface singularities; generic orbits; generic singularities; exceptional Lie algebras; Springer correspondence B. Fu, D. Juteau, P. Levy and E. Sommers, \textit{Generic singularities of nilpotent orbit closures}, [arXiv:1502.05770]. Singularities in algebraic geometry, Geometric invariant theory, Lie algebras of linear algebraic groups Generic singularities of nilpotent orbit closures | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The authors give a realization of Nakajima quiver varieties and of the action of the Weyl group on them using the notion of \((A,c)\)-complexes, a generalization of differential complexes which they develop in the paper. The case of affine quivers obtained by finite subgroups of \(\text{SL}(2,{\mathbb C})\) via the McKay correspondence and its relation to moduli of sheaves on the projective plane are discussed in detail.
More precisely, let \(\mathbf Q\) be a quiver, \(I\) the set of its vertices, \(({\mathbf v},{\mathbf w})=(\{v_a\}_{a\in I},\allowbreak\{w_a\}_{a\in I})\in {\mathbb N}^I\times {\mathbb N}^I\), \(\zeta=(\zeta_{\mathbb C},\zeta_{\mathbb R})\in {\mathbb C}^I\times {\mathbb R}^I\), and let \({\mathcal M}_\zeta({\mathbf v},{\mathbf w})\) be the quiver variety determined by these data [\textit{H.~Nakajima}, Duke Math. J. 76, No. 2, 365--416 (1994; Zbl 0826.17026)]. The authors consider a suitable quotient \(A\) of the path algebra of \(\mathbf Q\), and the graded \(A\)-module
\[
M=\bigoplus_a (V_a\otimes P_a)\oplus(W_a\otimes S_a[-1]),
\]
where \(V_a,W_a\) are \({\mathbb C}\)-vector spaces with \(\dim V_a=v_a\) and \(\dim W_a=w_a\), \(P_a\) is the indecomposable projective left \(A\)-module generated by the length zero path \((a)\in A\) and \(S_a\) is the simple left \(A\)-module given by the quotient of \(P_a\) by all paths of length greater than zero. The data \((B,i,j)\) in the definition of \({\mathcal M}_\zeta({\mathbf v},{\mathbf w})\) are shown to be equivalent to an \((A,c)\)-duplex structure on \(M\), i.e., to a degree 1 map \(d: M\to M\) such that \(d^2=c\), where \(c\) is a degree 2 central element of \(A\). Moreover, \(d^2\) and the \(d\)-Laplacian \((\sqrt{-1}/2)(dd^*+d^*d)\) are identified with the complex and the real moment map for the Nakajima quiver variety, respectively; the operator \(d^*\) in the definition of the \(d\)-Laplacian is the Hermitian adjoint of \(d\) with respect to the natural Hermitian structure on \(M\). For any \(\zeta_{\mathbb C},\zeta_{\mathbb R}\), this identification induces a bijection between the points of \({\mathcal M}_\zeta({\mathbf v},{\mathbf w})\) and isomorphism classes of Hermitian \((A,c)\)-duplex structures on \(M\) with \(d^2=\zeta_{\mathbb C}\) and \((\sqrt{-1}/2)(dd^*+d^*d)=\zeta_{\mathbb R}\).
Next, the authors describe an action of the Weyl group \(\mathbf W\) of \(\mathbf Q\) on the subspace of degree 2 central elements of \(A\) and show that, for generic \(c\), this action induces a functorial Weyl group action on the categories of \((A,c)\)-duplexes. More precisely, if \(s_a\) is the Weyl group element corresponding to the vertex \(a\) of \(\mathbf Q\), to \(s_a\) corresponds a natural functor \({\mathcal R}_a\) from the category of \((A,c)\)-duplexes to the category of \((A,s_a(c))\) duplexes. This functor induces a bijection \({\mathcal M}_\zeta({\mathbf v},{\mathbf w})\to {\mathcal M}_{s_a(\zeta)}(s_a({\mathbf v},{\mathbf w}))\), which is shown to coincide with the classical Weyl group action on Nakajima varieties [\textit{G.~Lusztig}, Ann. Inst. Fourier 50, No. 2, 461--489 (2000; Zbl 0958.20036); \textit{A.~Maffei}, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 1, No. 3, 649--686 (2002; Zbl 1143.14309); \textit{H.~Nakajima}, Math. Ann. 327, No. 4, 671--721 (2003; Zbl 1060.16017)].
In case \({\mathbf Q}\) is the quiver associated to a finite subgroup \(\Gamma\) of \(\text{SL}(2,{\mathbb C})\) via the McKay correspondence, the algebra \(A\) can be replaced by the Morita equivalent algebra \(A_\Gamma=\wedge\rho\otimes {\mathbb C}[\Gamma]\), where \(\rho\) is the natural 2-dimensional representation of \(\Gamma\). The authors are then able to reformulate the definition of \({\mathcal M}_\zeta({\mathbf v},{\mathbf w})\) entirely in terms of the representation theory of \(A_\Gamma\) and of \(\widetilde{A}_\Gamma=A_\Gamma\otimes {\mathbb C}[d]/(d^2-c)\). Finally, they show how this description is related to the moduli space of coherent sheaves on a noncommutative deformation of the projective plane [\textit{V.~Baranovsky, V.~Ginzburg} and \textit{A.~Kuznetsov}, Compos. Math. 134, No. 3, 283--318 (2002; Zbl 1048.14001)] and how to recover the classical description of Nakajima varieties as moduli spaces of torsion-free sheaves on \({\mathbb P}^2\) with fixed framing at infinity from the representation theory of \(\tilde{A}_\Gamma\). Nakajima quiver varieties; Weyl group; Koszul duality I. Frenkel, M. Khovanov, O. Schiffmann, Homological realization of Nakajima varieties and Weyl group actions, math.QA/0311485, 2003. Simple, semisimple, reductive (super)algebras, Representations of quivers and partially ordered sets, Algebraic moduli problems, moduli of vector bundles Homological realization of Nakajima varieties and Weyl group actions | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 about the action of \(K\) on \(\mathfrak p\), where \(\mathfrak g\) is a finite dimensional complex reductive Lie algebra, \(\theta\) is an involution of \(\mathfrak g\) with corresponding Cartan decomposition \(\mathfrak g = \mathfrak k \oplus \mathfrak p\); and \(K\) is the connected subgroup of the adjoint group \(G\) of \(\mathfrak g\) with Lie algebra \(\mathfrak k\). The \(K\)-sheets of \(\mathfrak p\) are the irreducible components of the subsets \(\mathfrak p^{(m)} := \{x\in \mathfrak p: \dim K\cdot x = m\}\), \(m\in \mathbb N\). Any \(K\)-sheet is a finite disjoint union of Jordan \(K\)-classes; here the Jordan \(K\)-class of \(x\in \mathfrak p\) is \(J_K(x) := \{y\in \mathfrak p:\) there exists \(k\in K, k\cdot\mathfrak p^x = \mathfrak p^y \}\). The notion of \textit{sheet} can be defined for an arbitrary representation of a reductive group [\textit{W. Borho} and \textit{H. Kraft}, Comment. Math. Helv. 54, 61--104 (1979; Zbl 0395.14013)]. The study of these varieties is related to various problems in Lie theory. A case of of particular importance is the adjoint action of a reductive group on a Lie algebra; here a fruitful approach to the study of sheets is through \textit{Slodowy slices}, see \S 7.4 in [\textit{P. Slodowy}, Simple singularities and simple algebraic groups. Lecture Notes in Mathematics. 815. Berlin-Heidelberg-New York: Springer-Verlag. (1980; Zbl 0441.14002)]. In the case under study of the action of \(K\) on \(\mathfrak p\), the author first considers the so-called group case (where \(\mathfrak g = \mathfrak g'\times \mathfrak g'\) and \(\theta\) is the flip) and obtains some results about Jordan classes. Next he proves smoothness of \(K\)-sheets for the classical types. A parametrization using generalized Slodowy slices is established under some technical conditions; it is shown that these conditions hold in type A, that is when \(\mathfrak g = \mathfrak{gl}_N\) or \(\mathfrak{sl}_N\). As a consequence, a complete description of the \(K\)-sheets, their dimensions and the determination of the Dixmier \(K\)-sheets (those that contain a semisimple element) and the rigid nilpotent \(K\)-orbits is achieved in type A. semisimple Lie algebra; Cartan decomposition; nilpotent orbit; Slodowy slice; sheet; Jordan class Bulois, M, Sheets of symmetric Lie algebras and slodowy slices, J. Lie Theory, 21, 1-54, (2011) Group actions on varieties or schemes (quotients), Simple, semisimple, reductive (super)algebras, Semisimple Lie groups and their representations Sheets of symmetric Lie algebras and Slodowy slices | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \(0\) and \(Z\) over \(k\) a normal surface. An order over \(Z\) is a coherent sheaf \(\mathcal{O}_X\) of \(\mathcal{O}_Z\)-algebras with generic fibre \(k(X):=\mathcal{O}_X\otimes k(Z)\), a central simple \(k(Z)\)-algebra. The order \(\mathcal{O}_X\) is considered as a non-commutative surface \(X\) with a finite map to the (commutative) surface \(Z\).
\textit{D. Chan} and \textit{C. Ingalls} generalized Mori's minimal model program to the case of orders over surfaces [Invent. Math. 161, No. 2, 427--452 (2005; Zbl 1078.14005)]. Canonical singularities of orders over surfaces are defined. They are non-commutative analogues of Kleinians singularities that arise naturally in the minimal model program. Canonical singularities of orders are classified using their minimal resolutions. They are explicitly described as invariant rings for the action of a finite group on a full matrix algebra over a regular local ring. It is proved that canonical singularities of orders are Gorenstein, their Auslander--Reiten quivers are described. A simple version of the McKay correspondence is given. canonical singularities; noncommutative surface; McKay correspondence; Gorenstein; Auslander-Reiten quiver Chan, D.; Hacking, P.; Ingalls, C., Canonical singularities of orders over surfaces, Proc. Lond. Math. Soc., 98, 83-111, (2009) Singularities in algebraic geometry, Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.), McKay correspondence Canonical singularities of orders over 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 In the paper under review, the author deals with the problem of classifying complete \(\mathbb{C}\)-subalgebras of \(\mathbb{C}[[t]]\), the ring of formal power series in one variable.
A discrete invariant providing such a classification is a semigroup \(\Gamma\), in \(\mathbb{N}\), obtained by taking the orders of the elements of a given \(\mathbb{C}\)-subalgebra of \(\mathbb{C}[[t]]\). Hence, the problem is reduced to classify complete \(\mathbb{C}\)-subalgebras of \(\mathbb{C}[[t]]\) with a given semigroup \(\Gamma\). So, it is possible to define the space \(R_{\Gamma}\) of all \(\mathbb{C}\)-subalgebras of \(\mathbb{C}[[t]]\) with given \(\Gamma\).
As semigroups arising from unibranch curve singularities are exactly the so-called numerical semigroups, the Author focuses on \(R_{\Gamma}\) for this type of semigroup and the study of the space \(R_{\Gamma}\) is motivated by showing how it relates to the Zariski moduli space of curve singularities on the one hand and to a moduli space of global singular curves on the other.
In particular, \(R_{\Gamma}\) is proved to be an affine variety by providing an algorithm, which yields its defining equations in an ambient affine space in terms of the given semigroup; some examples show how to use these results to explicitly compute \(R_{\Gamma}\). Moreover, the question is addressed of whether or not \(R_{\Gamma}\) can always be identified with an affine space and whether is there a numerical criterion for this; although the problem remains open, certain types of semigroups are identified, for which \(R_{\Gamma}\) is always an affine space, and for general \({\Gamma}\) a finite stratification of \(R_{\Gamma}\) is described, by locally closed subsets corresponding to subalgebras with a fixed number of generators. The relationship between \(R_{\Gamma}\) and the Zariski moduli space \(\mathcal{M}_{\Gamma}\) is also explored, by explicitly computing the natural map from \(R_{\Gamma}\) to \(\mathcal{M}_{\Gamma}\) in some special cases.
The same algebraic problem addressed in the present paper was considered (yet only in an abstract and scheme-theoretic perspective) by \textit{S. Ishii} [J. Algebra 67, 504--516 (1980; Zbl 0469.14013)]. curve singularities; classification; numerical semigroups; Zariski moduli space Families, moduli of curves (algebraic), Singularities of curves, local rings Classifying complete \(\mathbb{C}\)-subalgebras of \(\mathbb{C}[[t]]\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \textit{snc pair} $(X,\Delta)$ is given by a smooth variety $X$ over a field and a simple normal crossing divisor $\Delta = \sum D_i$ on $X$. The irreducible components of the intersections $D_{i_1} \cap \cdots \cap D_{i_r}$ are called \textit{strata} of the pair and the blow-up of a stratum is called \textit{toroidal}. For a snc pair $(X,\Delta)$, an ideal sheaf $J\subset \mathcal{O}_X$ is \textit{toroidally resolved} if the support of $\mathcal{O}_X / J$ does not contain any strata. \par The main result of the paper under review (cf. Theorem 10) shows that, given a snc pair $(X,\Delta)$ and an ideal sheaf $J\subset \mathcal{O}_X$, there exists a sequence of toroidal blow-ups which toroidally resolves $J$. This answers a question by S. Keel (cf. Question 2). The key ingredient in the proof is the construction of the \textit{toroidal hull} $J^t \supset J$ (cf. Definition-Theorem 17). This reduces the problem to the resolution of a toroidal ideal, which is known by the work of \textit{E. Bierstone} and \textit{P. D. Milman} in [J. Algebraic Geom. 15, No. 3, 443--486 (2006; Zbl 1120.14009)]. \par In Corollary 5, the author of the paper under review recovers a result by \textit{J. Tevelev} [Am. J. Math. 129, No. 4, 1087--1104 (2007; Zbl 1154.14039)]. resolution; toroidal variety Minimal model program (Mori theory, extremal rays), Birational geometry Partial resolution by toroidal blow-ups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper was written in the spring of 1978 and was based on two talks in the Séminaire sur les Singularités des Surfaces at the École Polytechnique in March of that year. The purpose of the talks was to give an introduction to the theory of polar cycles and polar classes of singular projective varieties, based on the author's paper in Ann. Sci. Éc. Norm. Supér., IV. Sér. 11, 247-276 (1978; Zbl 0401.14007), and then to compare these classes to the Chern-Mather classes and the Chern- MacPherson classes of the varieties in order to obtain enumerative information on the singularities. Thus the main new results presented in the talks were formulas for the Euler obstruction of an isolated hypersurface singularity and of ``ordinary singularities'', i.e., those that occur on generic projections of smooth varieties. In particular, it turned out that the local Euler obstruction of a pinch point of a surface in 3-space is equal to 1, which is also the Euler obstruction of a non- singular point.
At the same time, by completely different methods, a more general formula for the Euler obstruction was obtained by \textit{A. Dubson} [C. R. Acad. Sci., Paris, Sér. A 287, 237-240 (1978; Zbl 0387.14005)]; later, a formula for the local Euler obstruction in terms of Chern classes of vector bundles on the Nash transformation of the variety was obtained by Gonzalez-Sprinberg and Verdier [\textit{G. Gonzalez-Sprinberg}, Astérisque 82-83, 7-32 (1981; Zbl 0482.14003)]. The methods presented in the present paper have therefore become unnecessary in a certain sense - however, they do provide an example of how global classes may be used to compute local invariants of singularities. Chern-MacPherson classes; ordinary singularities; local Euler obstruction Piene, R., Cycles polaires et classes de Chern pour LES variétés projectives singulières, (Introduction à la théorie des singularités, II, Travaux en Cours, vol. 37, (1988), Hermann Paris), 7-34 Singularities in algebraic geometry, Classical real and complex (co)homology in algebraic geometry, Characteristic classes and numbers in differential topology, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Enumerative problems (combinatorial problems) in algebraic geometry Cycles polaires et classes de Chern pour les variétés projectives singulières. (On polar cycles and Chern classes of singular projective 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 a series of papers [Doc. Math., J. DMV 5, 553--594 (2000; Zbl 0971.14029); ``Global sections of line bundles on a wonderful compactification of the general linear group'', \url{arXiv:math/0305033}; J. Algebr. Geom. 14, No. 3, 439--480 (2005; Zbl 1080.14043)], \textit{I. Kausz} constructed a compactification \(\mathrm{KGL}_n\) of the general linear group \(\mathrm{GL}_n\), showed how the space of global sections of an arbitrary \(\mathrm{GL}_n \times \mathrm{GL}_n\)-linearized line bundle on \(\mathrm{KGL}_n\) and its orbit-closures decomposes into a direct sum of simple \(\mathrm{GL}_n \times \mathrm{GL}_n\)-modules, and proved a factorization theorem of generalized theta functions on the moduli stack of vector bundles on a curve.
The paper under review generalizes these results to the case of a symplectic group, working over an algebraically closed field of characteristic zero. The author defines a \textit{generalized symplectic isomorphism} of a vector space \(V\) of dimension \(2r\) with a symplectic form; and defines \(\mathrm{KSp}(V)\) to be the moduli space of generalized symplectic automorphisms of \(V\). There is an action of \(\mathrm{Sp}(V) \times \mathrm{Sp}(V)\) on \(\mathrm{KSp}(V)\) that extends the action on \(\mathrm{Sp}(V)\) defined by \((g_1,g_2).x := g_1 x g_2^{-1}\). \(\mathrm{KSp}(V)\) satisfies the following properties: {\parindent=6mm \begin{itemize}\item[-] \(\mathrm{KSp}(V)\) is smooth and the complement of \(\mathrm{Sp}(V)\) in \(\mathrm{KSp}(V)\) is a normal crossing divisor \(D_1 \cup \cdots \cup D_r\). \item[-] The \(D_i\) are smooth. \item[-] Every \(\mathrm{Sp}(V) \times \mathrm{Sp}(V)\)-orbit closure in \(\mathrm{KSp}(V)\) is a certain intersection of the \(D_i\). \item[-] For every point \(x \in \mathrm{KSp}(V)\), the normal space to the \(\mathrm{Sp}(V) \times \mathrm{Sp}(V)\)-orbit of \(x\) contains a dense orbit of the isotropy group \(\mathrm{Sp}(V)_x\). \item[-] \(\mathrm{KSp}(V)\) is a closed subvariety of \(\mathrm{KGL}_n\).
\end{itemize}} He also describes the intersections \(\bigcap_{i \in I} D_i\) for \(I \subset \{ 0,\dots,r-1 \}\). He then goes on to study \(\mathrm{Sp}(V) \times \mathrm{Sp}(V)\)-modules of the form \(\mathrm{H}^0(\mathrm{KSp}(V), \mathcal{O}(\sum a_i D_i))\); and he proves a decomposition result for this module. Finally, he proves a factorization theorem of generalized theta functions on the moduli stack of symplectic bundles. It must be mentioned that, in a later paper [J. Reine Angew. Math. 631, 181--220 (2009; Zbl 1177.14068)], the author uses the results in this paper to prove a strange duality theorem for the moduli space (or stack) of symplectic vector bundles on a curve. T. Abe, ``Compactification of the symplectic group via generalized symplectic isomorphisms'' in Higher Dimensional Algebraic Varieties and Vector Bundles , RIMS Kokyuroku Bessatsu B9 , Res. Inst. Math. Sci. (RIMS), Kyoto, 2008, 1-50. Classical groups (algebro-geometric aspects), Algebraic moduli problems, moduli of vector bundles Compactification of the symplectic group via generalized symplectic isomorphisms | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 survey some features of equivariant instanton partition functions of topological gauge theories on four and six dimensional toric Kähler varieties, and their geometric and algebraic counterparts in the enumerative problem of counting holomorphic curves. We discuss the relations of instanton counting to representations of affine Lie algebras in the four-dimensional case, and to Donaldson-Thomas theory for ideal sheaves on Calabi-Yau threefolds. For resolutions of toric singularities, an algebraic structure induced by a quiver determines the instanton moduli space through the McKay correspondence and its generalizations. The correspondence elucidates the realization of gauge theory partition functions as quasi-modular forms, and reformulates the computation of noncommutative Donaldson-Thomas invariants in terms of the enumeration of generalized instantons. New results include a general presentation of the partition functions on ALE spaces as affine characters, a rigorous treatment of equivariant partition functions on Hirzebruch surfaces, and a putative connection between the special McKay correspondence and instanton counting on Hirzebruch-Jung spaces. Donaldson-Thomas theory; BPS state counting; gauge theories on toric surfaces M. Cirafici and R. J. Szabo, Curve counting, instantons and McKay correspondences, \(J. Geom. Phys.\)72:54, 2013. arXiv:hep-th/1209.1486. Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Calabi-Yau theory (complex-analytic aspects), Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Calabi-Yau manifolds (algebro-geometric aspects) Curve counting, instantons and McKay correspondences | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The article under review focuses on an aspect of the mirror symmetry conjecture -- the relation between the stringy invariants of the pair of \(n\)-dimensional complex, possibly singular, Calabi-Yau varieties \((V,W)\):
\[
E_{\mathrm{st}}(V;u,v) = (-u)^nE_{\mathrm{st}}(W;u^{-1},v).
\]
The aims are to find combinatorial and representation-theoretic formulas for the stringy invariants and to give new pairs of Calabi-Yau orbifolds satisfying this relation.
The author considers varieties \(X\) and \(X^*\) which are hypersurfaces in toric varieties associated with dual reflexive lattice polytopes \(P\) and \(P^*\), as in the construction of \textit{V. Batyrev} and \textit{L. Borisov} [Invent. Math. 126, No. 1, 183--203 (1996; Zbl 0872.14035)]. The additional assumption is that \(P\) is invariant with respect to the linear action of a finite group \(\Gamma\) on the lattice \(M\) by representation \(\rho\), and that \(X\) and \(X^*\) are invariant with respect to the induced action. Then \textit{equivariant stringy invariants} \(E_{\mathrm{st},\Gamma}(X;u,v)\), which are polynomials in \(u,v\) with coefficients in the representation ring \(R(\Gamma)\), are defined. The main results of the paper is a general formula for the equivariant Hodge-Deligne polynomial of a non-degenerate hypersurface in a torus.
Theorem 4.10. This and the formula for the equivariant stringy invariant (given in Proposition 5.5) imply a representation-theoretic version of Batyrev-Borisov mirror symmetry, stated in terms of equivariant stringy invariants:
\[
E_{\mathrm{st},\Gamma}(X;u,v) = (-u)^{d-1}\det(\rho)\cdot E_{\mathrm{st},\Gamma}(X^*;u^{-1},v).
\]
The proofs of these results rely on earlier works of the author [Adv. Math. 226, No. 6, 5268--5297 (2011; Zbl 1223.14059)] and [Adv. Math. 226, No. 4, 3622--3654 (2011; Zbl 1218.52014)]; they are purely combinatorial.
It follows that by taking quotients of crepant resolutions \(\widetilde{X}/\Gamma\) and \(\widetilde{X}^*/\Gamma\) by \(\Gamma \in \mathrm{SL}(M)\) one obtains pairs of orbifolds with mirror Hodge diamonds, which do not appear in the Batyrev and Borisov's construction.
The last two sections of the paper are devoted to the study of special cases: the situation when \(P\) is a centrally symmetric reflexive polytope and the case of the Fermat quintic threefold. mirror symmetry; representation theory; hypersurfaces in toric varieties Stapledon, A, New Calabi-Yau orbifolds with mirror Hodge diamonds, Adv. Math., 230, 1557-1596, (2012) Calabi-Yau manifolds (algebro-geometric aspects), Mirror symmetry (algebro-geometric aspects) New Calabi-Yau orbifolds with mirror Hodge diamonds | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(f:(\mathbb C^n,0) \to \mathbb C \) be an isolated (\(n\)-dimensional) isolated singularity, that is a germ of an analytic function such that \(\nabla f := (f_{z_1}, \ldots, f_{z_n})\) vanishes at \(0\) but not for \(z\) near \(0\), \(z \not= 0\). An important number attached to \(f\) is its Łojasiewicz exponent \(\L _0(f)\). This can be defined as follows. Let \(B_0(f)\) be the set of positive real numbers \(\alpha\) such that \(\parallel \nabla (f) \parallel \leq C_{\alpha}{\parallel z \parallel}^{\alpha}\), for a suitable real number \(C_{\alpha} > 0\) and \(z\) close enough to \(0\). Then \({\L} _0(f) = \inf B_0(f)\). There are many equivalent definitions of this exponent. This article establishes an upper bound for \(\L _0(f)\), for singularities satisfying certain conditions. This bound uses elements that can be obtained from \(\Gamma (f)\), the set of compact faces of \({\Gamma} _{+}(f)\), the Newton diagram, or polyhedron, of \(f\).
The main result of this paper has to do with isolated singularities which are nondegenerate in the sense of Kouchnirenko (K-nondegenerate). This concept is defined in terms of solutions to certain auxiliary polynomial equations associated to each compact face \(S \in \Gamma (f)\). Moreover, the main theorem involves the notion of exceptional faces, namely faces in \(\Gamma (f)\) whose intersections with coordinate planes contain certain special segments. Let \(N(f)\) be the set of faces in \(\Gamma (f)\) which are not exceptional. Finally we need, for \(S\) as above, the real number \(\alpha(S)\), defined in terms of the intersections of the hyperplane spanned by \(S \in \Gamma (f)\) with the coordinate axes. The main theorem precisely says:
If \(f:(\mathbb C^n,0) \to \mathbb C \) is an isolated singularity, K-nondedgenerate, where \(N(f) \not= \emptyset\), then
\[
\L _0(f) \leq \max\big\{m(S): S \in N(f)\big\}\, .
\]
This result improves, when it can be applied, a similar inequality found by \textit{T. Fukui} [Proc. Am. Math. Soc. 112, No. 4, 1169--1183 (1991; Zbl 0737.58001)]; although Fukui's assumptions are weaker than Oleksik's. An example shows that sometimes the estimate of this paper is better than that of Fukui. Another example is given where in Oleksik's formula we have an equality, so that the given estimate cannot be improved. The paper concludes with a problem: can we give a suitable notion of an ``exceptional face'' so that, using it, the inequality of the main theorem becomes an equality? Łojasiewicz exponent; isolated singularity; Kouchnirenko's nondegeneracy; Newton diagram; exceptional face Oleksik, The Łojasiewicz exponent of nondegenerate singularities, Univ. Iagel. Acta Math. 47 pp 301-- (2009) Invariants of analytic local rings, Singularities in algebraic geometry The Łojasiewicz exponent of nondegenerate 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 From the authors' introduction: ``This article is part of the authors' program whose purpose is to prove the following conjecture on Resolution of Singularities of threefolds in mixed characteristic. The conjecture is a special case of Grothendieck's Resolution conjecture for quasi-excellent schemes.
Conjecture 1.1 Let \(C\) be an integral regular excellent curve with function field \(F\). Let \(S/F\) be a reduced algebraic projective surface and \(\mathcal X\) be a flat projective \(C\)-scheme with generic fiber \({\mathcal X}_F =S\). There exists a birational projective \(C\)-morphism \(\pi :{\mathcal Y} \to \mathcal X\) such that {\parindent=6mm \begin{itemize}\item[(i)] \({\mathcal Y}\) is everywhere regular. \item[(ii)] \(\pi^{-1}(\text{Reg} {\mathcal X}) \to \text{Reg} {\mathcal X} \) is an isomorphism.''
\end{itemize}} In a previous paper [J. Algebra 320, No. 3, 1051--1082 (2008; Zbl 1159.14009)], the authors have developed equicharacteristic techniques which extend to that situation. Using classical invariants introduced by Hironaka, they present a proof of the following:
Main Theorem 1.3. Let \((R, {\mathcal M}, k=k(x):= R/{\mathcal M} )\) be an excellent regular local ring of dimension four, \((Z,x):= (\text{Spec} R,{\mathcal M})\) and \((X,x):= (\text{Spec} R/(h),x)\) be a reduced hypersurface. Assume that the multiplicity \(m(x)\) of \((X,x)\) satisfies \(m(x) < p:=\text{char} k(x)\). Let \(v\) be a valuation of \(K(X)\) centered at \(x\). Then there exists a finite sequence of local blowing ups
\[
(X,x)=:(X_0,x_0) \leftarrow (X_1,x_1) \leftarrow \dots \leftarrow (X_n,x_n) ,
\]
where \(x_i\in X_i\), \(0\leq i \leq n\) is the center of \(v\), each blowing up center \(Y_i\subset X_i\) is permissible at \(x_i\) (in Hironaka's sense), such that \(x_n\) is regular.
The authors point out that the methods applied are global in nature and thus an extension of the main theorem to a global version should be possible. arithmetic varieties; Hironaka; resolution of singularities; blowing up; local uniformization V. Cossart, O. Piltant, Resolution of Singularities of Arithmetical Threefolds II. ArXiv e-prints, Dec. 2014. Valuations and their generalizations for commutative rings, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) Resolution of singularities of threefolds in mixed characteristic: case of small multiplicity | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be a finite subgroup of \(\text{SL}(n,\mathbb{C})\). \(G\) acts on \(\mathbb{C}^n\) freely outside a finite collection of linear subspaces of codimension \(\geq 2\). The \(G\)-Hilbert scheme \(\text{Hilb}^G(\mathbb{C}^n)\) parameterizing \(G\)-clusters on \(\mathbb{C}^n\) has been introduced by \textit{I. Nakamura} [J. Algebr. Geom. 10, No.4, 757--779 (2001; Zbl 1104.14003)] as a natural candidate to provide crepant resolutions of the quotient singularity \(\mathbb{C}^n/G\).\newline For \(n=2\), \(\text{Hilb}^G(\mathbb{C}^2)\) is the minimal resolution of \(\mathbb{C}^2/G\) [\textit{Y. Ito} and \textit{I. Nakamura}, Proc. Japan Acad., Ser. A 72, No.7, 135--138 (1996; Zbl 0881.14002)] and for \(n=3\), by the theorem of \textit{T. Bridgeland, A. King} and \textit{M. Reid} [J. Am. Math. Soc. 14, No. 3, 535--554 (2001; Zbl 0966.14028)], \(\text{Hilb}^G(\mathbb{C}^3)\) is smooth and is a crepant resolution of \(\mathbb{C}^3/G\). But for \(n\geq 4\), \(\text{Hilb}^G(\mathbb{C}^n)\) is not always smooth and the quotient \(\mathbb{C}^n/G\) might have no crepant resolution at all.\newline This paper is mostly concerned with the family of examples given by abelian groups:
\[
A_r(4)=\{g\in\text{SL}(4,\mathbb{C})\,| \,g \text{ diagonal}, g^{r+1}=1\},\quad r\geq 1.
\]
The main result is:
\(\text{Hilb}^{A_r(4)}(\mathbb{C}^4)\) is a smooth toric variety with canonical bundle \(\omega={\mathcal O}_{\text{Hilb}^{A_r(4)}(\mathbb{C}^4)}(\sum\limits_{k=1}^mE_k)\) with \(m=r(r+1)(r+2)/6\), where \(E_k\)'s are disjoint smooth exceptional divisors isomorphic to \(\mathbb{P}_1\times\mathbb{P}_1\times\mathbb{P}_1\). Blowing down \(E_k\) to some factors \(\mathbb{P}_1\times\mathbb{P}_1\) for each \(k\), it gives rise to crepant resolutions of \(\mathbb{C}^4/A_{r}(4)\), all of them differ by a sequence of flops of \(4\)-folds.
The result is proved in Section 4 (Theorem 4.1) by a deep study of the toric structure of \(\text{Hilb}^{A_r(4)}(\mathbb{C}^4)\) and Gröbner basis techniques. The easier special case \(r=1\) is treated in Section 3 (Theorem 3.5).\newline In Section 5, the authors compute a non-abelian case. Consider the alternating group \(\mathcal{A}_4\) acting by permutation on \(\mathbb{C}^4\) and restricted to \(\mathbb{C}^3\) considered as the standard representation. They apply their method to give a constructive proof of the known smooth and crepant structure of \(\text{Hilb}^{{\mathcal A}_4}(\mathbb{C}^3)\). The method could be further developed to investigate new higher-dimensional cases. Hilbert scheme of orbits; quotient singularities; 4-folds, toric geometry Parametrization (Chow and Hilbert schemes), \(4\)-folds, Group actions on varieties or schemes (quotients), Toric varieties, Newton polyhedra, Okounkov bodies, Representations of finite symmetric groups, Global theory and resolution of singularities (algebro-geometric aspects) On hypersurface quotient singularities of dimension \(4\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The aim of this paper is to show that the structures on \(K\)-theory used to formulate Lusztig's conjecture for subregular nilpotent \(\mathfrak{sl}_n\)-representations are, in fact, natural in the McKay correspondence. The main result is a categorification of these structures. The no-cycle algebra plays the special role of a bridge between complex geometry and representation theory in positive characteristic.
For Part I, cf. Compos. Math. 138, No. 3, 337--360 (2003; Zbl 1086.17010). Modular Lie (super)algebras, Singularities of surfaces or higher-dimensional varieties, Twisted and skew group rings, crossed products, Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects), Representation theory for linear algebraic groups Subregular representations of \(\mathfrak{sl}_n\) and simple singularities of type \(A_{n-1}\). 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 This is a lovely article: It answers a concrete, explicit mathematical problem by concrete results from abstract theory. The question is the following: ``Is it possible for the universal enveloping algebra of an infinite dimensional Lie algebra to be noetherian?'' The conjecture stated by the authors is then that a Lie algebra is finite dimensional if and only if the universal enveloping algebra \(U(L)\) is noetherian.
To help answering the question, and to strengthen the validity of the conjecture, the authors prove that the conjecture holds for the enveloping algebra \(U(W_+)\) of the positive Witt algebra \(W_+\), and as a consequence, for the full Witt algebra \(U(W)\). They also prove the conjecture true for the Virasoro algebra \(V\), and any infinite dimensional \(\mathbb Z\)-graded simple Lie algebra of polynomial growth. It is even so that all central factors of \(U(V)\) are non-noetherian. The authors use a very explicit representation of the Witt algebra by generators and relations. Then it is possible to give an explicitly given homomorphism \(\rho: U(W_+)\rightarrow K[t;\tau]\), which by definition makes the image \(R:=\text{im}(\rho)\) birationally commutative. Here \(K\) is a field and \(\tau\in\text{Aut}_k(K)\). Then one can use the classification of birationally commutative projective surfaces to prove that \(R\) is not noetherian.
The actual point schemes are not needed for the results of the article, but the homomorphism \(\rho\) is constructed using the truncated point schemes of \(U(W_+)\) which have geometric points parameterizing graded \(U(W_+)\)-modules with Hilbert series \(1+s+\dots+s^n\). As this is the classification of birationally commutative graded domains of Gelfand Kirillov dimension 3, it also follows that the GK dimension of \(R\) is 3, which is a nice bonus result of the article.
Every computation is explicitly given, and because Macaulay2 is used, however just to verify the computations, appendices with the scripts are given.
This article proves that the noncommutative moduli theory can be applied to explicit and concrete problems. birationally commutative algebra; centerless Virasoro algebra; infinite dimensional Lie algebra; non-noetherian universal enveloping algebra; Witt algebra; birationally commutative projective surfaces Sierra, S.S., Walton, Ch.: The universal enveloping algebra of Witt algebra is not noetherian. ArXiv:1304.0114 [math.RA] Noncommutative algebraic geometry, Universal enveloping algebras of Lie algebras, Rings arising from noncommutative algebraic geometry, Virasoro and related algebras The universal enveloping algebra of the Witt algebra is not Noetherian | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 representation theory of quivers, it is well-known that a finite connected quiver satisfies the trichotomy: of finite representation type, of tame representation type, or of wild representation type. In commutative algebra, the relevant problem concerns the complexity of maximal Cohen-Macaulay (MCM for short) modules, roughly speaking, concerns the dimension of the parameter family of rank \(r\) indecomposable MCM modules for each \(r>0\) over the given ring \(R\). Its geometric analogue is the case when \(R=k[X]\) is the coordinate ring of \(X\), where \(X \subset \mathbb{P}^n\) is a closed \(m\)-dimensional subvariety (or a subscheme) over a field \(k\). In the case, MCM modules correspond to arithmetically Cohen-Macaulay (ACM for short) sheaves, namely, locally Cohen-Macaulay sheaves without intermediate cohomology (with respect to the given embedding).
As a folklore, the families of indecomposable ACM sheaves also seems to enjoy the finite-tame-wild trichotomy. It is then interesting on which \(X\) is of finite, tame, or wild CM-representation type. CM-finite reduced projective ACM varieties are classified, they are either projective spaces, rational normal curves, smooth quadrics, the Veronese surface in \(\mathbb{P}^5\), or the cubic scroll in \(\mathbb{P}^4\) [\textit{D. Eisenbud} and \textit{J. Herzog}, Math. Ann. 280, No. 2, 347--352 (1988; Zbl.0616.13011)]. CM-tame varieties include CM-countable varieties (e.g. singular quadrics of corank \(1\)) and varieties where indecomposable ACM sheaves are parametrized by a curve (e.g. smooth elliptic curves).
The main goal of the paper under review is to classify all the reduced closed ACM subschemes \(X \subset \mathbb{P}^n\) of positive dimension by their CM-representation type, when the base field \(k\) is algebraically closed field of \(\mathrm{char}(k) \neq 2\). In particular, the authors bring the complete list of CM-finite and CM-tame reduced closed ACM subschemes, and show that all the others are of CM-wild. As a consequence, reduced ACM closed subschemes \(X\) of positive dimensions and the indecomposable ACM sheaves on \(X\) (up to isomorphism and degree shift) satisfy the finite-tame-wild trichotomy.
The key idea to show the CM-wildness, the authors develop an idea of \textit{Y. A. Drozd} and \textit{G.-M. Greuel} [J. Algebra 246, No. 1, 1--54 (2001; Zbl.1065.14041)]: they use a representation embedding from the category \(\textbf{Rep}_{\Upsilon}\) of finite-dimensional representations of a quiver \(\Upsilon\) to the category \(\textbf{MCM}_{k[X]}\) of MCM modules over \(k[X]\). If one can choose a quiver \(\Upsilon\) of wild representation type (e.g. Kronecker quivers with \(\ge 3\) arrows) and such an embedding, this will imply that \(X\) is CM-wild (Section 3 and Theorem A). To actually construct such a quiver and an embedding, one requires to find a pair of ACM sheaves good enough. The authors considered a general linear section \(Y \subset X\) of codimension \(c\), and taking the \(c\)-th syzygy module of an Ulrich module (roughly speaking, an MCM module of maximal number of minimal generators) on \(Y\) over \(X\). Except the case when \(X\) (and thus \(Y\)) is of minimal degree, this will reduce the problem to show the CM-wildness of \(Y\) (Section 4, 5 and Theorem B, C). Combining these data, except for the varieties of minimal degree, the Ulrich-wildness of a curve, or a surface linear section immediately implies the CM-wildness of \(X\) (Section 8, 9). For the varieties of minimal degree, one cannot reduce to the CM representation type of linear sections. The authors bring a method to deal with cones over smooth varieties of minimal degrees and directly determine their CM representation type (Section 7). algebraic geometry; commutative algebra; arithmetically Cohen-Macaulay sheaves; representation type Sheaves in algebraic geometry, Cohen-Macaulay modules, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Representation type (finite, tame, wild, etc.) of associative algebras The Cohen-Macaulay representation type of arithmetically Cohen-Macaulay 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 Denote by \(SU_X (r)\) the (coarse) moduli space of semistable rank-\(r\) vector bundles with trivial determinant line bundle over a compact Riemann surface \(X\) of genus \(g \geq 2\). It is known that the Picard group of \(SU_X (r)\) is freely generated by an ample line bundle \({\mathfrak L}\), which is called the determinant bundle on \(SU_X (r)\). For any integer \(k \geq 0\), the global sections of \({\mathfrak L}^{\otimes k}\) are sometimes called the generalized \(k\)-th order theta functions on \(SU_X (r)\), because they may be regarded as non-abelian generalizations of the classical theta functions associated with the Jacobian of a Riemann surface. During the past ten years, determinant bundles and their spaces of generalized theta functions have turned out to be fundamental objects in the construction and study of conformal quantum field theories, and some of the basic geometric properties of these mathematical objects (such as the famous Verlinde formulas) have been predicted by means of physical reasonings.
The present report, delivered by one of the most active contributors to these recent developments, provides a beautiful survey on the fundamental ideas, methods, and most important results in this fascinating area linking algebraic geometry and mathematical physics.
The author reviews, in ten brief sections, the moduli spaces \(SU_X (r)\), their determinant bundles and associated rational maps to projective spaces, the base loci of these rational maps, the particular case of semistable rank-2 vector bundles, the various proofs of the Verlinde formula for rank-2 bundles, the conjectural aspects of the so-called strange duality (which seems to be familiar to physicists), and the (again physically motivated) problem of describing the natural projective representation of the Teichmüller modular group \(\Gamma_g\) in \(\text{PGL} (H^0 (SU_X (r), {\mathfrak L}^{\otimes k})\).
The overview of the present state of knowledge is enhanced by the discussion of ten open problems in this context. These problems, among many others, form a crucial obstruction to the complete understanding of the analytic, geometric and physical nature of generalized theta functions. In this regard, the article under review may also be seen as a challenging proposal (or as a guiding program) for further research in this direction. generalized \(k\)-th order theta functions; moduli space of semistable rank-\(r\) vector bundles; determinant bundle; Jacobian of a Riemann surface; conformal quantum field theories; Verlinde formula; strange duality A. BEAUVILLE, \textit{Vector bundles on curves and generalized theta functions: recent results and open} \textit{problems}, Complex Algebraic Geometry, MSRI Publications 28 (1995), 17--33. Vector bundles on curves and their moduli, Theta functions and abelian varieties, Algebraic moduli problems, moduli of vector bundles, Theta functions and curves; Schottky problem, Two-dimensional field theories, conformal field theories, etc. in quantum mechanics Vector bundles on curves and generalized theta functions: Recent results and open problems | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This article can be regarded as the culmination of a deep, involved project of the authors, making significant progress in the understanding of Kac-Moody Lie algebras, cluster structures on Weyl group-indexed subgroups the of the unipotent group and the dual semicanonical basis. It is a wide generalisation of their earlier work on the Dynkin cases A, D and E [\textit{C. Geiss}, \textit{B. Leclerc} and \textit{J. Schröer}, Invent. Math. 165, No. 3, 589--632 (2006; Zbl 1167.16009)].
Fix a finite quiver \(Q\) with no oriented cycles. Let \(\Lambda=\Lambda_Q\) be the corresponding preprojective algebra (which may be infinite dimensional). Let \(W_Q\) be the Weyl group of \(Q\). For each element \(w\in W\), a Frobenius subcategory \(\mathcal{C}_w\) of the category \(nil(\Lambda)\) of all finite dimensional nilpotent representations of \(\Lambda\) was associated to \(w\) in [\textit{A. Buan, O. Iyama, I. Reiten} and \textit{J. Scott}, Compos. Math. 145, 1035--1079 (2009; Zbl 1181.18006)], where it was shown that the corresponding stable category, \(\underline{\mathcal{C}_w}\) is \(2\)-Calabi-Yau. These results were also discovered independently in [\textit{C. Geiss, B. Leclerc} and \textit{J. Schröer}, Cluster algebra structures and semicanonical bases for unipotent groups, Preprint \url{arXiv:math/0703039v4} [math.RT], (2007)], in the special case where \(w\) is adaptable. Each reduced expression \textbf{i} for \(w\) gives rise to corresponding maximal rigid \(\Lambda\)-module \(V_{\mathbf{i}}\).
A cluster algebra \(\mathcal{A}(\mathcal{C}_w)\) is associated to \(\mathcal{C}_w\), with initial seed given by \(V_{\mathbf{i}}\) (for any choice of reduced expression \textbf{i}). It is shown that \(\mathcal{A}(\mathcal{C}_w)\) has a natural realisation as a certain subalgebra of the graded dual of \(U(\mathbf{n})\), where \textbf{n} is the positive part of the symmetric Kac-Moody Lie algebra \textbf{g} of the same type as \(\Lambda\). It is shown that all the cluster monomials lie in the dual of Lusztig's semicanonical basis of \(U(\mathbf{n})\) and that the intersection of the dual semicanonical basis with \(\mathcal{A}(\mathcal{C}_w)\) is a basis for \(\mathcal{A}(\mathcal{C}_w)\). It is further shown that \(\mathcal{A}(\mathcal{C}_w)\) is isomorphic to the coordinate ring of of the finite dimensional unipotent subgroup \(N(w)\) (associated to \(w\)) of the symmetric Kac-Moody group attached to \textbf{g}.
It is also shown that inverting the generators of the coefficient ring gives rise to the algebra of regular functions on the unipotent cell associated to \(w\), solving a conjecture of \textit{A. Buan, O. Iyama, I. Reiten} and \textit{J. Scott} [loc. cit.].
Also interesting is that the endomorphism algebra of \(V_{\mathbf{i}}\) is shown to be quasihereditary, allowing a description of mutation of maximal rigid modules in \(\mathcal{C}_w\) in terms of \(\Delta\)-dimension vectors of corresponding modules over the endomorphism algebra.
Another spin-off is a new categorification of every acyclic cluster algebra with a skew-symmetric exchange matrix and a certain choice of coefficients.
This thorough, well-written paper covers a lot of ground and establishes very general strong, results, and is likely to be a standard reference in the future. Cluster algebra; Kac-Moody group; Kac-Moody algebra; Semicanonical basis; Frobenius category; Preprojective algebra; Representation theory; Unipotent group; Algebraic group; Calabi-Yau category; Quasihereditary algebra Geiß, C.; Leclerc, B.; Schröer, J., Kac-Moody groups and cluster algebras, Adv. Math., 228, 1, 329-433, (2011), MR2822235 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Cluster algebras, Special varieties, Representations of quivers and partially ordered sets, Universal enveloping (super)algebras, Quantum groups (quantized enveloping algebras) and related deformations, Representation theory for linear algebraic groups, Loop groups and related constructions, group-theoretic treatment, Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations Kac-Moody groups and cluster algebras | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This is Part II of the book, whose first Part has been reviewed in [Histoires hédonistes de groupes et géométries. Tome 1. Paris: Calvage et Mounet (2013; Zbl 1275.51001)]. Certainly that review is also in vogue in respect to Part II. In Part II, one can find a work-out of some topics of Part I, as well as some new topics. As such we mention, among others, Hecke algebras, Bruhat decomposition, Dynkin diagrams, Clifford chains, classical Lie groups, finite subgroups of \(\text{SO}(3)\) and \(\text{SO}(\mathbb{R})\) and its representations, McKay correspondence and characters of subgroups of \(\text{SU}_2(\mathbb{C})\).
Both the two parts give a beautiful view into group theory and geometry. actions of groups; linear algebra; topological groups; endomorphisms; Grassmannians; echelon matrices; groups preserving a bilinear form; quaternion fields; algebraic combinatorics; Lie groups; Platonic solids; topics from the projective plane; orthogonal groups; unitary groups; symplectic groups; Young tableaux; algebraic geometry; algebraic curves; surfaces configurations; special varieties; graphes; projective line; conics; representation theory; McKay correspondance Ph. Caldero, J. Germoni, \textit{Histoires Hédonistes de Groupes et de Géométries [Hedonistic Histories of Groups and Geometries].} Vol. 2, Calvage et Mounet, Paris, 2015. Introductory exposition (textbooks, tutorial papers, etc.) pertaining to geometry, Affine analytic geometry, Projective analytic geometry, Geometry of classical groups, Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations), Euclidean analytic geometry, Analytic geometry with other transformation groups, General theory of linear incidence geometry and projective geometries, Representations of finite symmetric groups, Finite automorphism groups of algebraic, geometric, or combinatorial structures, Curves in algebraic geometry, Singularities of curves, local rings Hedonistic histories of groups and geometries. Vol. 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 the paper under review, the author presents geometric representations of Heisenberg and Clifford algebras on the equivariant cohomology of the moduli space of framed torsion free sheaves on the complex projective plane \(\mathbf{C}P^2\), and on the equivariant cohomology of the Hilbert scheme of points on the resolution \(X_r\) of the simple singularity \(\mathbf{C}^2/\text\textbf{Z}_r\). The main ideas are to exhibit explicit correspondences inside products of equivariant subvarieties of the relevant spaces, and to use equivariant localization to prove that the correspondences satisfy the defining relations of Heisenberg and Clifford algebras.
\(\quad\)In Section 2 and 3, the author reviews basic facts about Heisenberg algebras, Clifford algebras and the boson-fermion correspondence. Quiver varieties were introduced in Sect.~4. Both the moduli space \(\mathcal M(r, n)\) of framed rank-\(r\) torsion free sheaves on \(\mathbf{C}P^2\) and the Hilbert scheme \((X_r)^{[n]}\) of points on the resolution \(X_r\) are special cases of quiver varieties. Torus actions on \(\mathcal M(r, n)\) and \((X_r)^{[n]}\) are defined in Section 5; the fixed loci are also identified. In Section 6, the author studies the equivariant cohomology of \(\mathcal M(r, n)\) and \((X_r)^{[n]}\), and localization of certain correspondences. Section 7 and 8 are devoted to geometric constructions of the Clifford algebras and the Heisenberg algebras respectively. In Section 9, it is proved that geometric constructions satisfy the defining relations of Heisenberg and Clifford algebras. In Section 10, the author describes a geometric interpretation of the boson-fermion correspondence. In Section 11, relations between the moduli spaces \(\mathcal M(r, n)\) and \((X_r)^{[n]}\) are investigated. framed sheaves; Hilbert schemes; Heisenberg and Clifford algebras A.M. Licata, Framed torsion-free sheaves on \$ \(\backslash\)mathbb\{C\}\{\(\backslash\)mathbb\{P\}\^\{\{2\}\}\} \$ , Hilbert schemes, and representations of infinite dimensional Lie algebras, Adv. Math. 226 (2011) 1057. Algebraic moduli problems, moduli of vector bundles, Parametrization (Chow and Hilbert schemes), Infinite-dimensional Lie (super)algebras Framed torsion-free sheaves on \(\mathbb {CP}^2\), Hilbert schemes, and representations of infinite dimensional Lie algebras | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(\Bbbk\) be an algebraically closed field of characteristic zero, and let \(\Gamma\) be an additive subgroup of \(\Bbbk\). Results of Kaplansky-Santharoubane and Su classify intermediate series representations of the \textit{generalised Witt algebra} \(W_{\Gamma}\) in terms of three families, one parameterised by \(\mathbb{A}^2\) and two by \(\mathbb{P}^1\). In this note, we use the first family to construct a homomorphism \(\Phi\) from the enveloping algebra \(U(W_{\Gamma})\) to a skew extension \(\Bbbk [\mathbb{A}^2] \rtimes \Gamma\) of the coordinate ring of \(\mathbb{A}^2\). We show that the image of \(\Phi\) is contained in a (double) idealizer subring of this skew extension and that the representation theory of idealizers explains the three families. We further show that the image of \(U(W_{\Gamma})\) under \(\Phi\) is not left or right noetherian, giving a new proof that \(U(W_{\Gamma})\) is not noetherian.
We construct \(\Phi\) as an application of a general technique to create ring homomorphisms from shift-invariant families of modules. Let \(G\) be an arbitrary group and let \(A\) be a \(G\)-graded ring. A graded \(A\)-module \(M\) is an \textit{intermediate series} module if \(M_g\) is one-dimensional for all \(g \in G\). Given a shift-invariant family of intermediate series \(A\)-modules parametrised by a scheme \(X\), we construct a homomorphism \(\Phi\) from \(A\) to a skew extension of \(\Bbbk [X]\). The kernel of \(\Phi\) consists of those elements which annihilate all modules in \(X\). generalised Witt algebra; intermediate series representation; idealizer Graded rings and modules (associative rings and algebras), Universal enveloping (super)algebras, Noncommutative algebraic geometry Generalised Witt algebras and idealizers | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper is devoted to a converse problem of Galois theory of differential fields. The questions connected with the classification of the subset of Picard-Vessiot extensions of a differential field \(F\) consisting of algebraic extensions are considered. The authors give an effective algorithm, allowing to build the so-called canonical differential equation for an algebraic extension of the field \(F\) with given Galois group \(G\) in the case, when \(F=Q(z)\). For the construction of such algorithm it is necessary to take into account the following circumstances.
An extension of \(F\) can be given by the differential equations of various orders.
\(G\) can have two exact not equivalent representations of identical dimension. That is an extension can be given by two not equivalent, in the sense of Levi's transformation, differential equations of identical order.
A differential equation of Fuchs class having only algebraic singularities, can have a transcendental solution.
The authors show, that the above mentioned difficulties can be overcome by limiting to three points of singularity \((0, 1, \infty)\) and considering the primitive subgroups \(SL(2)\) and \(SL(3)\) as \(G\). The canonical differential equation is searched by means of the method of undetermined coefficients. The exponents in the points of singularity are determined proceeding from consideration of the representations of \(G\) in the space of the holomorphic differential on the corresponding algebraic curve, Fuchs relation for exponents or the Weierstrass gap formula. For the calculation of the so-called accessory parameter or a condition of absent logarithmic ramification or a condition of presence of the rational solution at suitable tensor degrees of the required equation is used. For the eight finite primitive subgroups of \(SL(3)\), their computational results can be seen as an analog of Schwarz' list. differential fields; Galois theory Van der Put, M.; Ulmer, F., Differential equations and finite groups, \textit{Journal of Algebra}, 226, 2, 920-966, (2000) Differential algebra, Inverse Galois theory, Coverings of curves, fundamental group Differential equations and finite groups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(\mathcal F\) be a differential field of characteristic zero with derivation \(\delta\) and field of constants \(\mathcal C\). Assume that \(\mathcal F\) is differentially closed. Let \(\mathcal F\{y_1, \dots, y_N\}\) denote the ring of differential polynomials over \(\mathcal F\) (ordinary polynomials in indeterminates \(\delta^iy_j\) which are differentiated as the notation implies). A subset \(\Sigma\) of the affine space \(\mathcal F^N\) is closed in the \(\delta\) topology if it is the set of common zeros of a subset of \(\mathcal F\{y_1, \dots, y_N\}\). If \(\Sigma\) is irreducible and if \(I(\Sigma)\) is the ideal of differential polynomials vanishing on it, the absolute dimension \(a(\Sigma)\) is the transcendence degree of the quotient field of \(\mathcal F\{y_1, \dots, y_N\}/I(\Sigma)\). The notions of \(\delta\)-closed subset and absolute dimension generalize in an obvious way to arbitrary algebraic varieties over \(\mathcal F\).
The main result of this paper is the following theorem: Let \(A\) be an abelian variety over \(\mathcal F\) of dimension \(g\) which contains no abelian subvariety that descends to \(\mathcal C\). Let \(\Gamma \subset A\) be a \(\delta\)-closed subgroup of finite absolute dimension. Then any \(\delta\)-closed subset \(\Sigma\) of \(\Gamma\) Zariski dense in \(A\) has absolute dimension at least \(g+1\).
Let \(A^\#\) denote the \(\delta\)-closure of the torsion points of \(A\). As a consequence of the above theorem, and previous work of the first author, the authors obtain the following results of \textit{E. Hrushovski} and \textit{Z. Sokolovic} [``Minimal subsets of differential closed fields'', Trans. Am. Math. Soc. (to appear)]. Suppose \(A\) is simple and does not descend to \(\mathcal C\). Then:
Any proper \(\delta\)-closed subset of \(A^\#\) is finite;
any \(\delta\)-closed subset of \(A^\# \times A^\#\) is a finite union of translates of \(\delta\)-closed subgroups;
if \(B\) is another simple abelian variety over \(\mathcal F\) which does not descend to \(\mathcal C\) and is not isogenous to \(A\) then any proper \(\delta\)-closed subset of \(A^\# \times B^\#\) is a finite union of points or points times one of the factors.
The proof of the main result depends on the first author's earlier work on \(D\) schemes, and in particular on an analysis of the universal \(D\) schemes associated to \(\Gamma\) and \(\Sigma\). simple abelian variety over differential field; gap theorem; absolute dimension; universal schemes A. Buium and A. Pillay, A gap theorem for abelian varieties over differential fields, Math. Research Letters 4 (1997), 211-219. Algebraic theory of abelian varieties, Differential algebra, Transcendence (general theory) A gap theorem for abelian varieties over differential fields | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be a connected semi-simple complex Lie group, \(B\) its Borel subgroup, \(T\) a maximal complex torus contained in \(B\), and Lie \((T)\) its Lie algebra. This setup gives rise to two constructions; the generalized nonperiodic Toda lattice and the flag manifold \(G/B\). The Toda lattice for \((G,B,T)\) is the dynamical system on the cotangent bundle \(T^* \text{Lie}(T)\) endowed with the canonical holomorphic symplectic form and the holomorphic Hamiltonian function
\[
H(p,q)=(p,p)-\sum_{\text{simple roots } \alpha_i}(\alpha_i,\alpha_i) \exp\bigl(\alpha_i (q)\bigr),
\]
where \((,)\) is any fixed nonzero multiplication of the Killing form on each simple component of Lie \((G)\) and the simple roots are given by the roots of \(B\) with respect to \(T\). This system is known to he completely integrable [\textit{B. Kostant}, Sel. Math., New Ser. 2, 43--91 (1996; Zbl 0868.14024) and Adv. Math. 34, 195--338 (1979; Zbl 0433.22008)]. Therefore the variety defined by the ideal generated by the integrals of motions is the Lagrangian analytic submanifold of \(T^*\text{Lie} (T)\). On the other hand, for the flag manifold \(G/B\) we have the small quantum cohomology ring \(QH^*(G/B,\mathbb{C})\) which is generated by second cohomology classes and parameters. Denoting as \(q_i\) the coordinates of the parameter space \(H^2(G/B,\mathbb{C})\) defined by \(q_i(\sum a_jp_j)= \exp (a_i)\) (here \(p_j\) is the cohomology class corresponding to the fundamental weights), the author describes the ring structure of \(QH^* (G,B)\):
Theorem 1. The small quantum cohomology ring \(QH^*(G/B, \mathbb{C})\) is canonical isomorphic to \(\mathbb{C}[p_1,\dots, p_l,q_1, \dots, q_l]/I\), where \(I\) is the ideal generalized by the nonconstant complete integrals of motions of the Toda lattice for the Langlands-dual Lie group \((G^v,B^v,T^v)\) o \((G,B,T)\).
In fact the author proves more in this paper. Using the quantum hyperplane section principle it is possible to compute the virtual numbers of rational curves in Calabi-Yau 3-fold complete intersections in homogeneous spaces with the knowledge of the quantum \({\mathcal D}\)-module structure of the ambient spaces. The author shows that the \({\mathcal D}\)-module structure for \(G/B\) is governed by the conservation laws of quantum Toda lattices which are the quantizations of the Toda lattices and still integrable [\textit{B. Kostant}, Invent. Math. 48, 101--184 (1978; Zbl 0405.12013) and London Math. Soc. Lect. Note Ser. 34, 287--316 (1979; Zbl 0474.58010)], \textit{A. Reyman} and \textit{M. Semenov-Tiam-Shansky}, Invent. Math. 54, 81--100 (1979; Zbl 0403.58004)]. The Hamiltonian operator he considers is
\[
\widehat H=\Delta-\sum_{\text{simple roots }\alpha_i} (\alpha_i, \alpha_i) \exp \bigl(\alpha_i(q) \bigr),
\]
where \(\Delta\) is the Laplacian on Lie \(T\) associated with the invariant form \((,)\). Let \({\mathcal D}\) be the differential operator algebra over \(\mathbb{C}\), generated by \(\hbar\frac {\partial} {\partial t_i}\), multiplication by \(\hbar\) and \(\exp t_i\).
Theorem II. The quantum \({\mathcal D}\)-module of \(G/B\) is canonically isomorphic to \({\mathcal D}/{\mathcal I}\) where \({\mathcal I}\) is the left ideal generated by the nonconstant complete quantum integrals of motions of the quantum Toda lattice for the Langlands-dual Lie group \(G^\vee, B^\vee,T^\vee)\) of \((G,B,T)\). lagrangian submanifold; quantum \({\mathcal D}\) modules; mirror theorem; dynamical system; holomorphic hamiltonian function Kim, B., Quantum cohomology of flag manifolds \(G / B\) and quantum Toda lattices, Ann. of Math. (2), 149, 1, 129-148, (1999) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics) Quantum cohomology of flag manifolds \(G/B\) and quantum Toda lattices. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Consider a finite group \(G\subset \mathrm{SL}(2,\mathbb{C})\), the surface \(\mathbb{C}^2/G\) with a unique singular point \(0\) and the minimal resolution \(\pi: X\rightarrow \mathbb{C}^2/G\). It is well-known that the exceptional divisor \(E=\pi^{-1}(0)\) consists of a tree of \((-2)\)-curves whose dual graph is a Dynkin diagram of type ADE. The structure sheaf \(S_i=\mathcal{O}_{E_i}\) of any irreducible component \(E_i\) of \(E\) is known to be a \(2\)-spherical object, that is, for any \(i\) we have \(S_i\otimes \omega_X\cong S_i\) and \(\text{Hom}(S_i,S_i[k])=\mathbb{C}\) if \(k=0,2\) and \(0\) otherwise. Any spherical object gives rise to a spherical twist, an autoequivalence of the bounded derived category of coherent sheaves \(D^b(X)\) on \(X\), and it was shown by \textit{P. Seidel} and \textit{R. Thomas} [Duke Math. J. 108, No. 1, 37--108 (2001; Zbl 1092.14025)] that the spherical twists associated to the \(S_i\) satisfy the braid relations. In type A they also showed that this action of the braid group is faithful. The purpose of the paper under review is to prove the faithfulness in types ADE.
To do this, the Garside structure on braid groups is used in an essential way. Background concerning this and further generalities on braid groups are presented in Section 2. In Section 3 the authors recall some facts about spherical twists and prove their main result. The idea is to show that a braid group element is completely determined by the action of the corresponding twist on the direct sum of the \(S_i\). In the last section an application to spaces of stability conditions is presented. Namely, \textit{T. Bridgeland} has shown in [Int.\ Math.\ Res.\ Not.\ 2009, No.\ 21, 4142--4157 (2009; Zbl 1228.14012)] that a connected component of the stability manifold of a certain subcategory \(\mathcal{D}\) of \(D^b(X)\) is a covering of the space of regular orbits of the Weyl group corresponding to the singularity type. He also proved that if the braid group action is faithful, then the covering is in fact the universal one. Hence, the results of the paper under review show that we have a universal covering not only in type A but in types ADE. spherical objects; braid groups; stability conditions; Kleinian singularities; spherical twists C. Brav and H. Thomas, Braid groups and Kleinian singularities , preprint, [math.AG] Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Singularities in algebraic geometry, Braid groups; Artin groups Braid groups and Kleinian singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \({\mathcal A}_g\) denote the moduli space of principally polarized abelian varieties of dimension \(g\geq 2\) and \({\mathcal A}^{\text{per\,}f}_g\) its perfect cone compactification. It is well-known that the theta divisor of the intermediate Jacobian of a cubic threefold has unique triple point and this is necessarily an odd 2-division point. As a generalization the authors consider for any \(g\) the locus \(I^{(g)}\) of principally polarized abelian varieties of dimension \(g\) whose theta divisor is singular at some odd 2-divison point. It is of codimension \(g\) for \(g\leq 5\) and it is conjectured that this is true for any \(g\).
The first result of the paper is a computation of the class of \(I^{g}\) in the Chow group under the assumption of the conjecture. The proof uses the interpretation of the gradient of a theta function as a section of a vector bundle. Moreover for \(g\leq 5\) the class of the closure of \(I^{(g)}\) in \({\mathcal A}^{\text{per\,}f}_g\) as well as its projection to the tautological ring are computed. For \(g=5\) any indecomposable 5-dimensional ppav with a triple point on the theta divisor is in the closure of the locus \(IJ\) of intermediate Jacobians of cubic threefolds in \({\mathcal A}^{\text{per\,}f}\). Apart from \(IJ\) the space \(I^{(5)}\) contains second irreducible component, namely the \({\mathcal A}_1\otimes \theta^{(4)}_{\text{null}}\), where \(\theta^{(4)}_{\text{null}}\) denotes the theta-null divisor in \({\mathcal A}_4\). The classes of \({\mathcal A}_1\otimes \theta^{(4)}_{\text{null}}\) and its closure in \({\mathcal A}^{\text{per\,}f}_5\) are computed. This gives finally the projection of the class of the closure of \(IJ\) to the tautological ring. Some results on the geometry of the boundary of \(IJ\) in \({\mathcal A}^{\text{per\,}f}_5\) are concluded. intermediate Jacobians; moduli problems of abelian varieties S. Grushevsky and K. Hulek, The class of the locus of intermediate Jacobians of cubic threefolds, Invent. Math. 190 (2012), no. 1, 119-168. \(3\)-folds, Transcendental methods, Hodge theory (algebro-geometric aspects), Picard schemes, higher Jacobians, Algebraic moduli of abelian varieties, classification, Algebraic cycles The class of the locus of intermediate Jacobians of cubic threefolds | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 we can get the details of the paper reviewed above [\textit{D. Abramovich} and \textit{J. Wang}, Math. Res. Lett. 4, No. 2--3, 427--433 (1997; Zbl 0906.14005)]. The authors provide a new proof of Hironaka's well-known theorem on resolution of singularities as follows.
Theorem: Let \(X\) be a variety of finite type over an algebraically closed field \(k\) of characteristic 0, let \(Z\subset X\) be a proper closed subset. There exists a modification \(f:X_1\to X\), such that \(X_1\) is a quasi-projective nonsingular variety and \(Z_1= f^{-1} (Z)_{\text{red}}\) is a strict divisor of normal crossings.
Structure of the proof: (1) We choose a projection \(X\to P\) of relative dimension 1 and apply semistable reduction to obtain a model \(X'\to P'\) over a suitable Galois base change \(P'\to P\) with Galois group \(G\).
(2) We apply induction on the dimension to \(P\). We may assume that \(P\) is smooth, and the discriminant locus of \(X'/G\to P\) is a strict divisor of normal crossings.
(3) A few auxiliary blow-ups make the quotient \(X'/G\) toroidal.
(4) A theorem of \textit{G. Kempf}, \textit{F. Knudsen}, \textit{D. Mumford} and \textit{B. Saint-Donat} [``Toroidal embeddings'', Lect. Notes Math. 339 (1973; Zbl 0271.14017)] about toroidal resolutions finishes the argument. resolution of singularities; toroidal embedding; divisor of normal crossings [AJ]Abramovich, D. \&Jong, A.J. ed, Smoothness, semistability and toroidal geometry.J. Algebraic Geom., 6 (1997), 789--801. Global theory and resolution of singularities (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies Smoothness, semistability, and toroidal geometry | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be a complex orthogonal or symplectic group and \(\mathcal{M}(r, d, \alpha)\) the moduli space of stable parabolic \(G\)-bundles of rank \(r\), degree \(d\) and weight type \(\alpha\) over a compact Riemann surface \(X\) of genus \(g\geq2\). \textit{N. Hitchin} [Duke Math. J. 54, 91--114 (1987; Zbl 0627.14024)] discovered that the cotangent bundle of the moduli space of stable bundles on an algebraic curve is an algebraically completely integrable system fibered, over a space of invariant polynomials, either by a Jacobian or a Prym variety of spectral curves. The aim of this paper is to study the Hitchin fibers for \(\mathcal{M}(r, d, \alpha)\). The author proves that the Hitchin fibers for the moduli space of stable parabolic symplectic or orthogonal Higgs bundles on an algebraic curve are Prym varieties of the spectral curve with respect to an involution. In this context, Higgs fields are strongly parabolic, meaning that the Higgs field is nilpotent with respect to the flag. This paper is organized as follows: In Section 2, the author gives the necessary details regarding parabolic symplectic or orthogonal Higgs bundles and their moduli. Section 3 deals with a description of the Hitchin fibration and the spectral data. In Section 4, the author proves the main result for three different cases, i.e., for \(G=\mathrm{Sp}(2m, \mathbb{C}), \mathrm{SO}(2m, \mathbb{C})\) and \(\mathrm{SO}(2m +1, \mathbb{C})\). For \(G=\mathrm{Sp}(2m, \mathbb{C})\) or \(\mathrm{SO}(2m+1, C)\), fibers are Prym variety of the spectral curve with respect to an involution wih fixed points. For \(G=\mathrm{SO}(2m, \mathbb{C})\), the spectral curve is singular but the fibers are Prym variety of the desingularised spectral curve with respect to an involution without fixed points. integrable system; moduli space; parabolic bundle Families, moduli of curves (algebraic), Algebraic moduli problems, moduli of vector bundles, Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems, Jacobians, Prym varieties, Algebraic moduli of abelian varieties, classification Hitchin fibration on moduli of symplectic and orthogonal parabolic Higgs bundles | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The principal congruence subgroup \(\Gamma[2]\) is the kernel of reduction mod~\(2\) in the symplectic group \(\Gamma={\text{Sp}}(4,{\mathbb Z})\). This paper examines the structure of the rings of vector-valued modular forms, and of the module of cusp forms, for \(\Gamma[2]\) and the two intermediate groups \(\Gamma_1[2]= M\equiv \begin{pmatrix}1_2&\ast \cr 0& 1_2\end{pmatrix} \) and \(\Gamma_0[2]=\left\{ M\equiv \begin{pmatrix}\ast&\ast \cr 0& \ast \end{pmatrix}\right\}\). As \(\Gamma/\Gamma[2]= {\text{Sp}}(4,{\mathbb Z}/2)\) is isomorphic to the symmetric group \({\mathfrak S}_6\) it is possible to investigate the intermediate cases by studying the representation of \({\mathfrak S}_6\) on spaces of modular forms for \(\Gamma[2]\) once one has a good description of the latter.
For the special case of scalar-valued modular forms for \(\Gamma[2]\) the problems were all solved by Igusa, and the authors are able to describe the rings of scalar-valued modular forms for \(\Gamma_1[2]\) and \(\Gamma_0[2]\) by the above procedure, the latter having been already computed by a different method by Ibukiyama. These results are obtained in section~9 of the paper: the earlier sections are mostly brief and establish notation and essential facts about Siegel modular varieties, modular forms and theta series, mainly following Igusa. Using their results they also recover the surprising result of Mukai that the Satake compactification of the Siegel modular variety associated with \(\Gamma_1[2]\) is isomorphic to the Igusa quartic: surprising because the Igusa quartic is also a model for the variety associated with \(\Gamma[2]\). Mukai proved this result by geometric means: here it appears via the modular forms, and in an appendix to this paper he provides a minor correction concerning the action of the Fricke involution.
The rest of the paper is concerned with vector-valued forms. Now there is more to be done because the structure for \(\Gamma[2]\) was not previously known, so the authors have to construct modular forms as well as study the group actions. First they obtain some dimension formulas, extending previous work of Wakatsuki in a more or less routine way. Next they consider the part generated by Eisenstein series, and calculate the representation of \({\mathfrak S}_6\) there. This part assumes \textit{a priori} the truth of the conjectures in [\textit{J. Bergström} et al., Int. Math. Res. Not. 2008, Article ID rnn100, 20 p. (2008; Zbl 1210.11060)], but what comes next is a dimension formula for the full ring of modular forms for \(\Gamma_1[2]\) and this is nevertheless unconditional. They note some coincidences among the dimensions of some of the pieces and are able to explain two of them.
To construct vector-valued modular forms, i.e.\ to find generators, the authors use a variety of techniques. Two of them, Rankin-Cohen brackets and gradients of theta functions, are not quite standard and are explained in detail here.
The question of relations requires two kinds of input. By geometric methods one can obtain results about the vanishing loci of the modular forms and therefore their possible dependence. To establish that the generators and relations found are in fact to describe the ring one needs not only the dimension formulae but also some control over the expected number of relations, syzygies and so on. The authors prove a Castelnuovo-Mumford regularity result, which also gives bounds on the weights of the generators.
This is now enough information to establish the structure of the rings and certain modules in some detail. The last few sections give some examples of this. Siegel modular form; vector-valued modular form; modular forms of level 2; theta series F. Cléry, G. van der Geer, S. Grushevsky: \textsl{Siegel modular forms of genus \(2\) and level \(2\).} International Journal of Mathematics \textbf{26}, No. 5 (2015) 1550034 (51 pages). DOI 10.1142/S0129167X15500342; zbl 1344.14019; MR3345511; arxiv 1306.6018 Modular and Shimura varieties, Moduli, classification: analytic theory; relations with modular forms, Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms, Cohomology of arithmetic groups Siegel modular forms of genus 2 and level 2 | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The aim of the article is to interpret results of M. Atiyah on classification of vector bundles on an elliptic curve into the language of factors of automorphy. The paper provides some results which are used without proofs by authors, in particular in \textit{I. Burban} and \textit{B. Kreussler} [``Vector bundles on degenerations of elliptic curves and Yang-Baxter equations'', \url{arxiv:0708.1685}] and \textit{A. Polishchuk} and \textit{E. Zaslow} [Adv. Theor. Math. Phys. 2, No. 2, 443--470 (1998; Zbl 0947.14017)], with proofs.
Let \(\Gamma\) be a group acting upon a complex manifold \(Y\). The \(r\)-dimensional factor of automorphy is a holomorphic function \(f: \Gamma \times Y \to \mathrm{GL}_r(\mathbb C)\) such that \(f(\lambda \mu, y)=f(\lambda, \mu y)f(\mu,y)\). Two factors of automorphy \(f\) and \(f'\) are said to be equivalent if there is a holomorphic function \(h: Y \to \mathrm{GL}_r(\mathbb C)\) such that \(h(\lambda y ) f(\lambda,y)=f'(\lambda,y) h(y).\)
Let \(X\) be a complex manifold, \(p: Y\to X\) its universal covering and \(\Gamma\) be the fundamental group of \(X\) acting on \(Y\) by deck transformations. Then there is a bijection between the set of equivalence classes of \(r\)-dimensional factors of automorphy and the set of isomorphism classes of vector bundles on \(X\) with trivial pull-back along \(p\).
The correspondence between 1-dimensional complex tori and elliptic curves and between classifications of holomorphic vector bundles on complex projective variety and of algebraic vector bundles on the same variety yield in the reformulation of Atiyah's results in terms of factors of automorphy.
The article consists of 5 sections. Section 1 is an introduction. Section 2 contains the construction of the correspondence between vector bundles of rank \(r\) and \(r\)-dimensional factors of automorphy. Section 3 is devoted to properties of factors of automorphy. In particular, their relation to theta functions is investigated as well as relation to operations on vector bundles. Section 4 is focused on study of complex tori. Also, the author proves a criterion when the factor of automorphy \(f\) defines a trivial bundle and criteria for two factors of automorphy to be equivalent. Section 5 deals with a classification of vector bundles over a complex torus.
The author works with factors of automorphy depending only on direction \(\tau\) of lattice \({\mathbb Z}+\tau{\mathbb Z}\) of a torus, i.e., with holomorphic functions \({\mathbb C}^{\ast}\to \mathrm{GL}_r({\mathbb C})\). elliptic curve; factor of authomorphy; Theta function; vector bundles; complex tori O. Iena, Vector bundles on elliptic curves and factors of automorphy, arXiv:1009.3230. Vector bundles on curves and their moduli, Elliptic curves, Algebraic moduli problems, moduli of vector bundles, Theta functions and abelian varieties Vector bundles on elliptic curves and factors of automorphy | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Mirror symmetry is a phenomenon discovered by physicists and used by mathematicians apriori as a tool for making predictions on enumerative invariants of Calabi-Yau varieties. However, in the last thirty years, apart from verifying such predictions, several mathematical techniques have been developed by pure mathematicians to investigate relationships between complex and symplectic geometries of Calabi-Yau varieties that are manifested by mirror symmetry.
One of the most celebrated conjectures in mirror symmetry, due to \textit{A. Strominger} et al. [Nucl. Phys., B 479, No. 1--2, 243--259 (1996; Zbl 0896.14024)], relates the complex geometry of a Calabi-Yau threefold with integral affine geometry on a three-dimensional real manifold \(B\). In rough terms, the manifestation of mirror symmetry in this context is that mirror pairs of Calabi-Yau varieties \((X,\breve{X})\) admit dual special Lagrangian torus fibrations \(f\colon X\to B\), and \(\breve{f} \colon \breve{X}\to B\). with singular fibers over a discriminant locus \(\Delta \subset B\). The mirror to \(X\), in the sense of Strominger-Yau-Zaslow (SYZ), can be obtained by first constructing a semi-flat mirror to the restriction of \(X\) to \(B\setminus \Delta\) obtained by dualizing the non-singular torus fibers and then compactifying it by considering appropriate corrections of the complex structure. These corrections are expected to be captured by counts of holomorphic discs in \(X\) with boundaries on torus fibers [\textit{K. Fukaya}, Proc. Symp. Pure Math. 73, 205--278 (2005; Zbl 1085.53080)]. However, constructing the mirror to a Calabi-Yau following this strategy is a notoriously difficult problem, in particular, as showing the existence of special Lagrangian fibrations is difficult. Moreover, it is technically challenging to provide a precise description of the counts of holomorphic disks with boundaries on the torus fibers.
Nonetheless, following early insights of Kontsevich-Soibelman [\textit{M. Kontsevich} and \textit{Y. Soibelman}, Prog. Math. 244, 321--385 (2006; Zbl 1114.14027)], Gross-Siebert show that algebro-geometric analogues of such counts of holomorphic disks can be recovered from the combinatorics of a normal crossing limit \(X_0\) of a large complex structure degeneration \((X_t)\) of \(X\). In particular, they reduce the problem to determine such counts to a combinatorial problem encoded on a \textit{canonical wall structure} on the affine manifold \(B\) which is viewed as a tropical limit of \(X_0\). The construction of the canonical wall structure in this paper, builds on ideas of the Gross-Siebert program, which the authors have been developing since early 2000's.
The Gross-Siebert program provides the most general mirror construction algebro-geometrically. In particular, this approach allows one to overcome the aforomentioned challenges in the context of SYZ mirror symmetry. Here, one does not need the special Lagrangian fibration \(X \rightarrow B\), but only the base \(B\) with a structure of integral affine manifold with singularities. Further, the theory of algebro-geometric counter-parts of the holomorphic discs appearing in SYZ mirror symmetry has been developed by the authors and their collaborators recently. These counterparts are given in terms of rational curves in the total space of the family \((X_t)\) with tangency conditions along the normal crossings divisor defined by the special fiber \(X_0\). Such counts are examples of so-called punctured Gromov-Witten invariants of \textit{D. Abramovich} et al. [``Punctured logarithmic maps'', Preprint, \url{arXiv:2009.07720}].
Building on punctured Gromov-Witten theory, Gross and Siebert recently provided a general mirror construction, known as ``intrinsic mirror symmetry'' [\textit{M. Gross} and \textit{B. Siebert}, Proc. Symp. Pure Math. 97, 199--230 (2018; Zbl 1448.14039)], in which the homogeneous coordinate ring of the mirror is directly constructed in terms of an explicit linear basis of so-called ``theta functions'', and where the structure constants of the product are given by punctured Gromov-Witten invariants. Whereas the ``intrinsic mirror symmetry'' construction is very direct, it is a bit far from the original SYZ picture. In the current paper, Gross and Siebert use punctured Gromov-Witten theory to give an algebro-geometric construction of the ``canonical wall structure'', which is the expected combinatorial structure on the base \(B\) of the SYZ fibration defined by the counts of Maslov index zero holomorphic disks. Moreover, they show that the mirror which can be constructed from this wall structure agrees with the mirror obtained by intrinsic mirror symmetry.
Technically, the contributions of the paper are as follows. Given a maximal log Calabi-Yau variety or a large complex structure degeneration of Calabi-Yau varieties satisfying some technical conditions, the authors in this article provide:
\begin{enumerate}
\item The definition of the canonical wall structure using punctured Gromov-Witten invariants,
\item The definition of logarithmic theta functions using punctured Gromov-Witten invariants,
\item The proof that the logarithmic theta functions agree with combinatorially defined theta functions constructed from the canonical wall structure (Theorem A),
\item The proof that the canonical wall structure is consistent (Theorem B), and;
\item The proof that the mirror constructed from the canonical wall structure agrees with the mirror constructed by intrinsic mirror symmetry (Theorem C).
\end{enumerate}
The proof of Theorem A uses the gluing formula for punctured Gromov-Witten invariants recently proved by \textit{Y. Wu} [``Splitting of Gromov-Witten invariants with toric gluing strata'', Preprint, \url{arXiv:2103.14780}]. The proof of Theorem B uses Theorem A and a version of deformation invariance in punctured Gromov-Witten theory, and more precisely follows from an explicit rational equivalence between cycles. Finaly, the proof of Theorem C also relies on the gluing formula for punctured Gromov-Witten invariants.
This paper symmetry is a key contribution in mirror symmetry, as it provides a general algebro-geometric mirror construction in the spirit of the more than 20 years old SYZ conjecture. Explicit examples of this construction building on this paper have recently been worked out by \textit{H. Argüz} and \textit{M. Gross} [Geom. Topol. 26, No. 5, 2135--2235 (2022; Zbl 07632769)]. mirror symmetry Mirror symmetry (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Logarithmic algebraic geometry, log schemes The canonical wall structure and intrinsic mirror symmetry | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(F\) be the algebraic closure of the finite field with \(q\) elements and let \(\mathcal Q\) be a quiver with the underlying graph of Dynkin type \(A_n\). The group \(G_d=\text{GL}_{d_1}(F)\times\cdots\times \text{GL}_{d_n}\) acts by conjugation on \(E_d=\bigoplus_{i\to j\in{\mathcal Q}}\Hom_F(F^{d_i},F^{d_j})\). In the paper under review the authors describe the \(G_d\)-orbits \(\mathcal O\) with the property that the orbit closure \(\overline{\mathcal O}\) (in the Zariski topology) is rationally smooth. The approach is to consider the corresponding quantized enveloping algebra and to study the action of the bar involution on PBW bases. Then the authors use Ringel's Hall algebra approach to quantized enveloping algebras and Auslander-Reiten quivers, and describe the commutation relations between root vectors. As a result they obtain explicit formulas for the multiplication of an element of PBW bases adapted to a quiver with a root vector as well as recursive formulas to study the bar involution on PBW bases. As a consequence the authors derive that if the orbit closure is rationally smooth, then it is smooth.
The recent paper [\textit{P. Caldero} and \textit{R. Schiffler}, Ann. Inst. Fourier 54, No. 2, 265--275 (2004; Zbl 1126.17013)] contains the characterization of the rationally smooth orbit closures of representations of quivers of type \(A\), \(D\) or \(E\). Comparing the methods of both papers, the present paper has the advantage that the approach is very explicit and recursive, and may be used for computer programming. But it cannot easily be generalized to type \(D\) and \(E\). representations of quivers; varieties of representations; rational smoothness; quantum groups; quantized enveloping algebra Robert Bédard and Ralf Schiffler, Rational smoothness of varieties of representations for quivers of type \({A}\), preprint. Quantum groups (quantized enveloping algebras) and related deformations, Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects), Singularities in algebraic geometry, Representations of quivers and partially ordered sets, Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers Rational smoothness of varieties of representations for quivers of type \(A\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be a finite subgroup of \(\text{SL}(n, \mathbb C)\), \(S_G\) the covariant algebra of \(G\) and \(\langle S_G\rangle_i\) the subspace of \(S_G\) of homogeneous degree \(i\). For each irreducible representation \(\rho\) of \(G\) let \(\langle \rho, (S_G)_i\rangle _G\) be the multiplicity of \(\rho\) in \((S_G)_i.\) The Molian series \(P_{S_{G, \rho}} (t)\) of \(S_G\) for \(\rho\) is defined by
\[
P_{S_{G, \rho}}(t)=\sum\langle \rho, (S_G)_i\rangle_G t^i.
\]
Explicit formulas for the Molian series are given in the case that \(G\) is one of the exceptional finite subgroups of \(\text{SL}(3, \mathbb C)\). Let \(\text{Hilb}^G(\mathbb C^n)\) be the universal subscheme of the Hilbert scheme \(\text{Hilb}^{| G| } (\mathbb C^n)\) parameterizing all smoothable scheme theoretic \(G\)-orbits of length \(| G| \). \(\text{Hilb}^G(\mathbb C^3)\) is studied, especially the fiber \(\pi^{-1}(0)\) of the Hilbert-Chow morphism \(\pi: \text{Hilb}^G(\mathbb C^3) \to \mathbb C^3/G\) in case that \(G\) is a finite subgroup of SO(3). An SO(3)-version of the McKay correspondence similar to the SU(2) case is given. Hilbert scheme; invariant theory; McKay quiver; McKay correspondence Gomi, Y; Nakamura, I; Shinoda, K, Coinvariant algebras of finite subgroups of SL\_{}\{3\}\(\mathbb{C}\), Can. J. Math., 56, 495-528, (2004) Group actions on varieties or schemes (quotients), Linear boundary value problems for ordinary differential equations, Parametrization (Chow and Hilbert schemes), Singularities of surfaces or higher-dimensional varieties Coinvariant algebras of finite subgroups of \(\text{SL} (3,\mathbb{C})\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The study of singular points of algebraic curves in the complex plane has a long history. Its beginnings can be traced back to Sir Isaac Newton, and the algebraic geometers of the nineteenth and early twentieth century developed it into a fascinating, already remarkably rich theory. One of the major achievements, during this period, was the resolution of singularities of such curves initiated by Max Noether.
From the 1920s on, the then new topological methods were applied to the local study of singularities of curves, knots and links. In the second half of the twentieth century, the newly developing singularity theory in higher dimensions also propelled the study of the singular points of plane curves, and the developments in this area have been tremendous since the late 1960s. In the course of its long history, singularity theory of plane curves has grown into a meeting point for many different disciplines of mathematics, including algebra, complex analysis, algebraic geometry, topology, and combinatorics. The interaction between ideas, methods and techniques from these various sources makes the study of singularities of plane curves particularly fascinating, enlightening, abundant and fruitful. Moreover, this subject provides a beautiful testing ground for geometric ideas, in general, and a perfect topic for developing a profound understanding of the principles of modern geometry, likewise.
The book under review, written by one of the leading experts in singularity theory, is a highly welcome attempt to present a systematic, comprehensive, versatile and up-to-date account of the present state of art of this venerable area within mathematics. Based on an M.Sc. course taught a number of times (since 1975) at the University of Liverpool, it has partly the character of an introductory textbook, and can be used as such, but it also discusses more recent, advanced and intradisciplinary topics from the forefront of current research in the singularity theory of plane curves. Thus the text, consisting of eleven chapters, is virtually divided into two main parts.
The first five chapters are kept to the level of the underlying M.Sc. course and, therefore, are more introductory and elementary in nature. They are meant to form the core of the book, providing the foundations of the classical theory of plane curve singularities. As for this part, the author has chosen the concept of equisingularity, i.e., the most important equivalence relation for singularities, as the general leitmotif for his approach. Equisingularity can be characterized from numerous different points of view, and the development of the distinct ideas and methods leading to that same concept is taken as the frame for an introduction to curve singularities. This is the mean feature of this approach, and of the book as a whole, that the author emphasizes the equivalence of differing concepts and methods from the beginning on, thereby demonstrating their appearance and power in an integrated account.
Chapter 1 compiles the necessary preliminary material: the definition of algebraic curves in the plane, intersection numbers, resultants and discriminants, manifolds and the implicit function theorem, polar curves and inflection points. All this is treated as basically familiar background material and not covered in every detail. The story starts with Chapter 2, where parametrizations of curves via Puiseux power series, branches of curves, multiplicities and tangent lines to curves are discussed. This is used in Chapter 3 to describe the resolution of curve singularities, including the blow-up process, the notion of infinitely near points, invariants of singularities, and the graph-theoretic interpretation of the configurations arising in the resolution process. Chapter 4 deals with the theory of contact of two branches of a curve, the Eggers tree associated with a branch, computing intersection numbers for curves with several branches, and the equivalent characterizations of the concept of equisingularity in the whole framework developed so far. Chapter 5 turns to the topological aspects of curve singularities, with a special emphasis on knots, links and the classical Alexander polynomial. Equisingularity is then reconsidered from this topological point of view.
The second part of the book, which comprises the remaining six chapters, is written at a more sophisticated level, gives introductions to a number of topics of current research, and even offers several new results of the author. In these more advanced chapters, the topological aspects of curve singularities play a dominant role.
Chapter 6 is devoted to the Milnor fibration, Milnor numbers, and the Euler characteristic of a fibration. The latter is used for several instructive calculations of Milnor numbers. Chapter 7 is entitled ``Projective curves and their duals''. The author gives proofs of the general Plücker theorems for singular plane curves, treats Klein's equation by using Euler characteristics of constructible functions, analyzes the singularities of a dual curve, and surveys some known results about curves with so-called maximal singularities.
The following three chapters are very up-to-date and lead up to the calculation of the monodromy of the Milnor fibration. Chapter 8 introduces calculations and notation for later use, including several numerical invariants of singularities and their representation using exceptional cycles on resolution trees. This chapter also contains an introduction to the topological zeta function à la Denef-Loeser.
Chapter 9 discusses the application of W. Thurston's decomposition theorems for 3-manifolds and for homeomorphisms of surfaces to the Milnor fibration. This chapter offers a novel view to the topology of curve singularities, with a number of results published for the first time. Among other things, the author presents a finiteness criterion for the monodromy and a close relation between the Eggers tree, the resolution graph and the Eisenbud-Neumann diagram of a singularity. Chapter 10 continues with new results in calculating the monodromy, mainly by using Seifert matrices and (again), the Thurston decomposition theorems. In addition, it is shown how to classify Seifert forms over a field, and how some of the numerical invariants required for the classification over the rational number field can be calculated.
The final Chapter 11 touches upon the more algebraic aspects of curve singularities. Ideals in the local ring of a singularity are related to the exceptional cycles studied earlier, and a link between ideals and Enriques's clusters of infinitely near points is established in form of a Galois correspondence. The discussion concludes with brief treatments of jets, equisingularity classes and the determinacy of functions.
Each chapter of the book comes with a section on ``Notes'' and ``Exercises''. The notes include historical remarks, references, comments on related material not covered in the text, and some hints for further research. The exercises form a balanced mixture of routine problems on applying the results in the text to concrete examples, on the one hand, and more challenging problems related to an alternative approach to a topic treated before, on the other. Also, the entire text is relaxed by numerous illustrating and instructive examples, and the bibliography is with more than 200 references more than ample.
All in all, this book, being partly an introductory textbook and partly an advanced research monograph, is extremely comprehensive, profuse and versatile. It contains a wealth of information, both classical and topical, on the attractive and evergreen area of plane algebraic curves, and it offers a lot of new insights to all kinds of readers.
The text reflects the author's great expertise in the field in a masterly way, and that just as much as his passion for the subject and his cultured attitude. His style of writing mathematics is utmost pleasant, nowhere formal, very user-friendly, throughout motivating and highly inspiring. No doubt, this book will quickly become a widely used standard text on singularities of plane curves, and a valuable reference book, too. textbook; algebraic curves; singularities; resolution of singularities; topology of singularities; monodromy C. T. C. Wall, \textit{Singular Points of Plane Curves}, Cambridge University Press, New York, 2004. Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Singularities in algebraic geometry, Singularities of curves, local rings, Plane and space curves, Global theory and resolution of singularities (algebro-geometric aspects), Monodromy on manifolds Singular points of plane curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(\overline{\mathcal M}_{g,n}\) be the Deligne-Mumford moduli space of stable curves of genus \(g\) with \(n\) marked points. Consider a a simple complex, finite-dimensional, Lie algebra \(\mathfrak g\), a positive integer \(\ell\) (the level) and an \(n\)-tuple of dominant integral weights \((\lambda_1, \dots, \lambda_n)\) with level of \(\lambda_i \leq \ell\) for all \(i\). The Wess-Zumino-Witten model of conformal field theory can be thought of as defining vector bundles on \(\overline{\mathcal M}_{g,n}\) whose fibers are the so-called vector spaces of conformal blocks, with ranks computed by the Verlinde formula. In this paper the authors consider \(\overline{\mathcal M}_{0,n}\), with \(\mathfrak g = \mathfrak{sl}_n\), \(\ell = 1\) and \(n\)-tuples \(\omega = (\omega_{j_1}, \dots, \omega_{j_n})\), where \(\omega_1, \dots, \omega_{n-1}\) are the fundamental weights. The divisor associated to the corresponding conformal block bundle is denoted \(D^n_{1, \omega}\), or simply \(D^n_{1, j}\) when \(\omega = (\omega_{j}, \dots, \omega_{j})\) is \(\mathbb S_n\)-invariant. The authors study these divisors. It is shown that the divisors \(D^n_{1, j}\) for various \(j =2, \dots, [\frac n2]\) span extremal rays of the cone of symmetric nef divisors. Evidence pointing to the validity of this result was first obtained in the following way: recurrence formulae for the Chern classes of conformal block divisors were given in [\textit{N. Fakhruddin}, Contemp. Math. 564, 145--176 (2012; Zbl 1244.14007)]. These formulae were implemented in a package of Macaulay and subsequent experimentation suggested the above mentioned fact. The authors also show that the morphisms from \(\overline{\mathcal M}_{0,n}\) to projective varieties defined by the divisors \(D^n_{1, \omega}\) factor through natural birational contractions defined in [\textit{B. Hassett}, Adv. Math. 173, No. 2, 316--352 (2003; Zbl 1072.14014)]. conformal blocks; Verlinde formula; nef divisors Maxim Arap, Angela Gibney, James Stankewicz, and David Swinarski, \?\?_{\?} level 1 conformal blocks divisors on \overline\?_{0,\?}, Int. Math. Res. Not. IMRN 7 (2012), 1634 -- 1680. Families, moduli of curves (algebraic), Two-dimensional field theories, conformal field theories, etc. in quantum mechanics, Geometric invariant theory \(sl_{n}\) level 1 conformal blocks divisors on \(\overline{M_{0,n}}\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \({\mathfrak O}\) be an order of a quadratic field of discriminant \(D\) and fix an integer \(N\geq 1\). We denote by \(\text{Pic}^+({\mathfrak O})\) the abelian group of invertible fractional \({\mathfrak O}\)-ideals modulo the principal ones defined by generators of positive norm. In this paper the authors define a Heegner triplet of type \((N,D)\) as a triplet \(({\mathfrak O},{\mathfrak n},[{\mathfrak a}])\), where \({\mathfrak n}\) is a primitive \({\mathfrak O}\)-ideal of norm \(N\) and \([{\mathfrak a}]\) an element of \(\text{Pic}^+({\mathfrak O})\), and prove that this concept generalizes the concept of Heegner point on the modular curve \(X_0(N)\). Let \({\mathfrak H}(N,D)\) be the set of integral quadratic forms of type \(aNX^2+ bXY+ cY^2\), with \(D= b^2-4Nac\), \(\gcd(aN,b,c)= \gcd(a,b,Nc)=1\). Furthermore, the authors prove that the group
\[
\Gamma_0(N)= \left\{ \left[ \begin{smallmatrix} \alpha&\beta\\ \gamma&\delta \end{smallmatrix} \right]\in \text{SL}(2,\mathbb{Z})\mid \gamma\equiv 0\pmod N\right\}
\]
operates on \({\mathfrak H}(N,D)\), the quotient \(H(N,D)= {\mathfrak H}(N,D)/ \Gamma_0(N)\) is a finite set and there is a bijection between \(H(N,D)\) and the set of pairs \(({\mathfrak n},[{\mathfrak a}])\). Moreover, when \(D<0\), they obtain a formula for the number of elements of \(H(N,D)\) in terms of the number of representations by genera of binary quadratic forms of discriminant \(D\). Heegner triplet; modular curve; integral quadratic forms; number of representations; genera of binary quadratic forms Arithmetic aspects of modular and Shimura varieties, General binary quadratic forms, Global ground fields in algebraic geometry Heegner points on modular curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(k=\overline k\) be an algebraically closed field; any non-hyperelliptic smoooth projective curve \(C\) of genus \(g\geq 3\) can be embedded into \(\mathbb{P}_k^{g-1}\) via its canonical map \(\phi\) (defined by its canonical divisor \(K_C\)), and its image \(\phi (C)\) is called a canonical curve. Its homogeneous coordinate ring \(k[x_0,\dots,x_{g-1}]/I_{\phi(C)}\) is Gorenstein, hence for any two generic linear forms \(h_1,h_2 \in k[x_0,\dots,x_{g-1}]_1\), the quotient ring \(k[x_0,\dots,x_{g-1}]/(I_{\phi(C)}\) is an Artinian Gorenstein ring, wirh Hilbert function: \((1,g-2,g-2,1)\).
Those rings have been widely studied, and in the case \(k\cong \mathbb{C}\), there is a bijective correspondence between such rings and forms of degree 3 in \(k[y_0,\dots,y_{g-2}]\), while if \(k\) has positive characteristics, the forms of degree 3 must be taken in the divided powers \(k\)-algebra on \(y_0,\dots,y_{g-2}\). Hence it is quite natural to try to relate geometric properties of \(\phi (C)\) and algebraic properties of the Artinian Gorenstein Algebra above.
One first result, which was already known for \(k\cong \mathbb{C}\), but it is proved here for any \(k=\overline k\), is that s smooth non-hyperelliptic projective curve \(C\) is either trigonal or isomorphic to a plane quintic if anf only if the corresponding cubic polynomial in \(y_0,\dots,y_{g-2}\) is a Fermat cubic \(y_0^3+\dots+y_{g-2}^3\), up to the natural action of \(GL_{g-2}\).
A new result here describes what happen when the cubic form corresponding to \(\phi (C)\) (in any characteristics) can be minimally written as a sum of \(g-1\) cubes of linear forms; in this case (here \(g\geq 5\)), \(C\) is either bielliptic, or isomorphic to a plane sextic with at most \(10-g\) double points as singularities, or to a complete intersection of a quadric and a quartic surface in \(\mathbb {P}^3\). Canonical curves; apolarity; Artinian Gorenstein Algebra; bielliptic curves E. Ballico, G. Casnati, and R. Notari, Canonical curves with low apolarity, J. Algebra 332 (2011), 229 -- 243. Special divisors on curves (gonality, Brill-Noether theory), Special algebraic curves and curves of low genus, Plane and space curves, Elliptic curves, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Projective techniques in algebraic geometry Canonical curves with low apolarity | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The authors continue their study [Am. J. Math. 112, No. 6, 1003-1071 (1990; Zbl 0734.14010)] of the \(\Gamma_ 0(p)\)-level structure on principally polarized abelian varieties: the algebraic stack \({\mathcal A}_ 2(\Gamma_ 0(p))\) classifying certain isogenies between abelian varieties is normal, four-dimensional, flat and Cohen-Macaulay over \(\mathbb{Z}_{(p)}\) having only isolated singularities. In that paper the property of being Cohen-Macaulay is deduced from a thorough analysis of an algebra \(R_ 2\) given explicitly by generators and relations which arises by studying the local moduli space of \({\mathcal A}_ 2 (\Gamma_ 0(p))\) at geometric points.
Using ideas from Hodge algebras and methods of commutative algebra the authors succeed to show, that in general, the ring \(R_ n\) is reduced, Cohen-Macaulay with \(n+1\) components consequently the ``worst'' singularities of \({\mathcal A}_ n (\Gamma_ 0(p))\) are normal and Cohen- Macaulay. This result forces \({\mathcal A}_ n (\Gamma_ 0(p))\) to be itself Cohen-Macaulay for \(p>2\) taking into accout a local fibration structure of the local moduli space. As the authors point out the statement is true for \(p=2\) as well. The paper stops by the words of the authors with two exercises in the deformation theory of abelian varieties. singularities of level structure; Cohen-Macaulay singularities; isogenies between abelian varieties; local moduli space; Hodge algebras C.L. Chai and P. Norman , '' Singularities of the \Gamma 0(p)-level structure '', J. of Algebraic Geometry 1, 2 (1992) 251- 277. Singularities in algebraic geometry, Isogeny Singularities of the \(\Gamma_ 0(p)\)-level structure | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 classification theorem for complex finite dimensional simple Lie algebras, and a classification of all complex finite dimensional simple Lie superalgebras. The last objects fits into two disjoint families: The ones of classical type, that is strict super-analogues of simple, finite dimensional, complex Lie algebras and the ones of Cartan type, which are simple but infinite dimensional. This classification result can be used as a basis for the classification, existence, and construction of simple Lie supergroups and simple algebraic supergroups. The problems above for Lie supergroups are solved by a superanalogue of Lie's Third Theorem, the problems in the algebraic situation remains.
In the standard situation, a constructive procedure giving all finite dimensional connected simple algebraic groups was given by Chevalley. Starting with a (complex) f.d. simple Lie algebra \(\mathfrak g\), a faithful \(\mathfrak g\)-module \(V\), a group of requested type is eventually constructed as a subgroup of \(\text{GL}(V)\). In particular, applying this construction gives all connected algebraic groups whose tangent Lie algebra is finite dimensional and simple. This method can be extended to the framework of reductive \(\mathbb Z\)-group schemes. By analogy, Chevalley's method is used in known work to the finite dimensional simple Lie superalgebras of classical type, to construct connected, algebraic supergroup-schemes (over \(\mathbb Z\)) which integrate such Lie superalgebras. In this article the author implements Chevalley's idea to simple Lie superalgebras of Cartan type. The main result is a constructive procedure for connected, algebraic supergroup-schemes (over any ring) whose tangent Lie superalgebra is simple of Cartan type. A seceond result in the article is a uniqueness theorem for algebraic supergroups given above.
The author starts of with a finite dimensional Lie superalgebra \(\mathfrak g\) of Cartan type. Then a detailed description of the root spaces with respect to a fixed Cartan subalgebra is given and the key notion of \textit{Chevalley basis} is introduced. The existence of such a Chevalley basis is proved, and a PBW-theorem for the Kostant \(\mathbb Z\)-form of the universal enveloping superalgebra of \(\mathfrak g\) is given.
The next step is to choose a faithful \(\mathfrak g\)-module \(V\), and to show that there is a lattice \(M\) in \(V\) fixed by the Kostant superalgebra and by a certain integral form \(\mathfrak g_V\) of \(\mathfrak g\). Let \((\text{salg}_{\Bbbk}\) be the category of commutative \(\Bbbk\)-superalgebras, and consider the functor \(G_V:(\text{salg}_{\Bbbk})\rightarrow(\text{groups})\) given as follows: For \(A\in(\text{salg}_{\Bbbk})\), \(G_V(A)\) is the subgroup of \(\text{GL}(A\otimes_{\mathbb Z} M)\) generated by homogeneous one-parameter subgroups associated with the root vectors and with the toral elements in a Chevalley basis. Finally, \(\mathbb G_V\) is the sheafification of \(G_V\).
The commutation relations among the generators gives a factorization of \(\mathbb G_V\) into a direct product of representable algebraic superschemes. Then \(\mathbb G_V\) is itself representable, which defines it as an \textit{affine} algebraic supergroup. This construction of \(\mathbb G_V\) yields an existence theorem of a supergroup having \(\mathfrak g_V\) as tangent Lie superalgebra. Also, the author proves the corresponding uniqueness theorem, that every such supergroup is isomorphic to some \(\mathbb G_V\).
As an example, the author constructs \(\mathbb G_V\) for \(\mathfrak g\) of type \(W(n)\) and \(V\) the defining representation; that is the Grassmann algebra with \(n\) odd indeterminates, and \(W(n)\) the algebra of superderivations.
The article is self-contained and detailed. It starts with all the necessary preliminaries on superalgebras, superspaces, and supergroups. It defines Lie superalgebras and functions with values in Lie superalgebras, and it gives the definition and properties of the two types of Lie superalgebras: Cartan type and classical type. For the first type, \(W(n)=\text{Der}_{\mathbb K}(\Lambda(n))\), the set of \(\mathbb K\)-(super)derivations of \(\Lambda(n)\), is important, and its Lie structure and its properties are given. Other necessary concepts are Cartan subalgebras, \(\mathfrak g\)-modules, roots, and root spaces; all are thoroughly defined and introduced. After the introductory chapters follow the integral structures with the main objective: To extend the classical notion of Chevalley bases for (semi)-simple Lie algebras. The Kostant superalgebra is defined, and explicit computations are used to prove results needed in the rest of the text, in particular in the proof of Kostant's PBW theorem: The Kostant superalgebra \(K_{\mathbb Z}(\mathfrak g)\) is a free \(\mathbb Z\)-module, that is, for any given total order \(\preceq\) of the set \(\tilde{\Delta}\coprod\{1,\dots,n\}\) a \(\mathbb Z\)-basis of \(K_{\mathbb Z}(\mathfrak g)\) is the set \(\mathcal B\) of ordered PBW-like monomials, i.e. all products without repetitions of factors of type \(X_{\tilde\alpha}^{\ell_{\tilde\alpha}},(H_i,n_i),X_{\tilde\gamma}\) with \(\tilde\alpha\in\tilde\Delta_{\overline 0}\), \(i\in\{1,\dots,n\}\), \(\tilde\gamma\in\tilde\Delta_{\overline 1}\), and \(\ell_{\tilde\alpha}\), \(n_i\in\mathbb N\) - taken in the right order with respect to \(\preceq\).
Algebraic supergroups \(G_V\) of Cartan type are defined and generalized from the classical situation. This gives a sheaf functor \(\mathbb G_V:(\text{salg})_{\Bbbk}\rightarrow(\text{groups})\) which is a sheaf functor such that \(\mathbb G_V(A)=G_V(A)\) when \(A\in(\text{salg})_{\Bbbk}\). These functors are then eventually proved to be the affine algebraic supergroups of Cartan type, the main object of interest in the rest of this article: Let \(\mathbb G_V\) be the affine supergroup of Cartan type built upon \(\mathfrak g\) and the \(\mathfrak g\)-module \(V\). Then \(\text{Lie}(\mathbb G_V)\) is quasi-representable, and actually representable, namely \(\text{Lie}(\mathbb G_V)=\mathcal L_{\mathfrak g_{V,\Bbbk}}\) as functors from \((\text{salg})_{\Bbbk}\) to \((\text{Lie}_{\Bbbk})\). The final chapter gives even more detailed results on the classical case.
The article is a good, advanced and technical, but self-contained study of the subject of Lie superalgebras. There is a lot of interesting and important generalizations, and the article, which is an attribute Pierre Cartier on his 80th birthday, is well worth working through. Lie supergroups; algebraic supergroups; algebraic supergroup-schemes; root spaces; Cartan subalgebra; Chevalley basis; universal enveloping superalgebra; superderivations; Kostant superalgebra Gavarini F., Algebraic supergroups of Cartan type, Forum Math. 26 (2014), no. 5, 1473-1564. Supervarieties, Noncommutative algebraic geometry, Simple, semisimple, reductive (super)algebras Algebraic supergroups of Cartan 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 We prove that a pair of singularities related by a transformation arising from the McKay correspondence are orbifold equivalent. From this we deduce a McKay type category equivalence for the matrix factorization categories. isolated singularities; Gorenstein singularities; matrix factorizations; crepant resolution; orbifold equivalence; McKay correspondence; Fourier-Mukai transform; conformal field theory correspondence; Landau-Ginzburg models; triangulated categories; quantum dimension Singularities in algebraic geometry, McKay correspondence, 2-categories, bicategories, double categories McKay correspondence and orbifold equivalence | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 beautifully illustrated paper, the authors address an open question of [\textit{M. Blickle}, J. Algebr. Geom. 22, No. 1, 49--83 (2013; Zbl 1271.13009)]: Is the set of \(F\)-pure Cartier submodules of the triple \((R,\phi, \mathfrak{a}^t)\) finite? The authors attack this question in the toric setting. Before addressing the main result, ample background on \(\phi\)-compatible and \(\phi\)-fixed ideals is presented and \(F\)-pure Cartier submodules are presented as a generalization of these notions. In particular, the finiteness of \(\phi\)-compatible ideals with respect to a Frobenius splitting and the finiteness of \(\phi\)-fixed ideals with respect to any Frobenius operator is known to be true. The authors remark that the finiteness result for \(\phi\)-fixed ideals does not hold on the set of fractional ideals, with a nice example which coincidentally is not toric. Setting \(\mathcal{F}\) to be the set of faces of \(t\) times the Newton polyhedron of \(\mathfrak{a}\), the authors show that the set of \(F\)-pure Cartier submodules of the triple \((R,\phi, \mathfrak{a}^t)\) corresponds to the ideals which are generated by the monomials in the relative interior of a face contained in a sub complex of \(\mathcal{F}\). The finite set of ideals in the above result are shown to be precisely the set of intermediate adjoint ideals (Definition 4.1) related to a toric log resolution and an effective divisor contained in the union of the support of an effective \(\mathbb{Q}\)-divisor, the singularities of the toric variety and the support of the monomial ideal \(\mathfrak{a}\). Frobenius splitting; Cartier algebra; toric variety; log resolution DOI: 10.1090/S0002-9947-2013-05856-4 Toric varieties, Newton polyhedra, Okounkov bodies, Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Multiplier ideals, Singularities in algebraic geometry Cartier modules on toric 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 \textit{J.-F. Boutot} proved the following result [Invent. Math. 88, 65--68 (1987; Zbl 0619.14029)]:
Theorem 1. Let \(k\) be a field of characteristic zero, and \(A\) and \(S\) be \(k\)-algebras essentially of finite type. If \(A\) is a pure subalgebra of \(S\) and \(S\) has at most rational singularities, then \(A\) has at most rational singularities.
Corollary 1. Let \(k\) be a field of characteristic zero, \(G\) a linearly reductive group over \(k\), and \(S\) a \(G\)-algebra of finite type which has at most rational singularities. Then, \(S^G\) has at most rational singularities.
In this paper, we discuss a reductive group analogue of corollary 1. As a reductive group is linearly reductive in the case of characteristic zero, the problem is for positive characteristics. \textit{R. Fedder} and \textit{K. Watanabe} [in: Commutative Algebra, Proc. Microprogram, Berkeley 1989, Publ., Math. Sci. Res. Inst. 15, 227--245 (1989; Zbl 0738.13004)] introduced the notion of F- rationality of rings of characteristic \(p>0\), utilizing the notion of tight closures by M. Hoechster and C. Huneke. Since then, it has been clarified that some of the definitions and results on singularities in characteristic zero are deeply connected to tight closures and Frobenius splittings in characteristic \(p\). In fact, the notion of F-rationality is a right characteristic \(p\) version of that of rational singularity. Nevertheless, an obvious interpretation of theorem 1 into the F-rational version is false even for \(\mathbb{G}_m\) [\textit{K. Watanabe}, J. Pure Appl. Algebra 122, 323--328 (1997; Zbl 0897.13027)]. Our main theorem is as follows.
Theorem 6. Let \(k\) be an algebraically closed field of characteristic \(p>0\), and \(G\) a connected reductive group over \(k\). Let \(V\) be a finite dimensional \(G\)-module, and set \(S:=\text{Sym}\,V\). If \(S\) is a good \(G\)-module, then \(S^G\) is strongly F-regular.
The notion of strong F-regularity was originated by \textit{M. Hochster} and \textit{C. Huneke} [J. Am. Math. Soc. 3, 31--116 (1990; Zbl 0701.13002)] in the line of studying tight closures. It is defined for noetherian rings of characteristic \(p>0\), and is stronger than F-rationality.
Good modules (or modules with good filtration) are important objects in representation theory of reductive groups. The theory of good modules is somewhat characteristic-free, and even base ring free in some sense. In general, the invariant subring \(S^G\) depends on the characteristic in a chaotic way. Roughly speaking, we only treat something similar to the rings of invariants in characteristic zero, imposing the condition \(S\) being good. On the other hand, the situation is not quite the same as in the case of characteristic zero. Even if \(S=\text{Sym}\,V\) is good, \(S^G\) may not be a pure subalgebra of \(S\). Although the assumption in theorem 6 (in characteristic \(p>0)\) is much stronger than that of corrollary 1 (in characteristic 0), the conclusion is also a little bit stronger than corollary 1. We also prove the following characteristic zero theorem, which looks more analogous to theorem 6.
Proposition 2. Let \(k\) be a field of characteristic zero, \(G\) a linearly reductive group over \(k\), and \(S\) a \(G\)-algebra of finite type. Assume that \(S\) is a UFD, and \(S\) has at most rational singularities. Then, we have \(S^G\) is of strongly F-regular type. tight closure; F-regularity; good module Hashimoto, M, \textit{good filtrations of symmetric algebras and strong F-regularity of invariant subrings}, Math. Z., 236, 605-623, (2001) Actions of groups on commutative rings; invariant theory, Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Group actions on varieties or schemes (quotients), Linkage, complete intersections and determinantal ideals Good filtrations of symmetric algebras and strong F-regularity of invariant subrings. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper is a substantial advancement in the homological version of the minimal model program. Let \(Y\) be a singular algebraic variety. The author constructs a categorical resolution \(\widetilde{D}\) of the category \(D^b(Y)\) for every Lefschetz decomposition of the category \(D^b(\widetilde{Z})\), where \(\widetilde{Z}\) is the exceptional divisor in a resolution of singularities \(\widetilde{Y}\to Y\). Then the author gives the conditions on the Lefschetz decomposition for \(\widetilde{D}\) to be a noncommutative resolution of \(D^b(Y)\), and for \(\widetilde{D}\) to be a crepant resolution [a development of \textit{D. Kaledin}, J. Geom. Funct. Anal. 17, No.~6, 1968--2004 (2008; Zbl 1149.14009), \textit{M. Van den Bergh}, J. Duke Math. J. 122, No. 3, 423--455 (2004; Zbl 1074.14013), \textit{M. Van den Bergh}, The legacy of Niels Henrik Abel, Berlin: Springer, 749--770 (2004; Zbl 1082.14005)].
The author also develops the conjecture of \textit{A. Bondal} and \textit{D. Orlov}, [Proceedings of the international congress of mathematicians, ICM 2002, Beijing, China, Vol. II, 47--56 (2002; Zbl 0996.18007)] that every singularity admits a minimal categorical resolution. In the author's interpretation, this conjecture splits into two: first, that \(D^b(X)\) for smooth \(X\) admits (multiple) minimal Lefschetz decompositions (conjecture 4.11) and second, that minimal Lefschetz decompositions give minimal categorical resolutions (conjecture 4.9). derived categories of coherent sheaves; categorical resolutions of singularities; noncommutative resolutions of singularities; crepant resolutions of singularities; Lefschetz decompositions Auslander, M.: Isolated singularities and existence of almost split sequences. In: Proc. ICRA IV, Lecture Notes in Mathematics, vol. 1178, pp.~194-241, Springer (1986) Derived categories, triangulated categories, Global theory and resolution of singularities (algebro-geometric aspects), Minimal model program (Mori theory, extremal rays) Lefschetz decompositions and categorical resolutions of singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper under review splits into two parts: The first one, the arithmetic part deals with the classification of triples (L,\(\phi\),(S,\(\theta)\)), where L is a unimodular lattice, i.e. a non- degenerate integral symmetric bilinear form, \(\phi\) is an involution of L and S is a possibly degenerate sublattice of L with an involution \(\theta\) given on it such that \(\phi\) induces \(\theta\) on S. Firstly a system of invariants of such triples is given for the genus and secondly - using the well known results of Eichler and Kneser on the classes in the genus of indefinite lattices - all isomorphism classes of such triples with even L are enumerated under some mild restrictions.
In the second part arithmetic is applied to geometry. To any even- dimensional real nonsingular projective algebraic variety A there is associated an involution namely complex conjugation on the homology lattice L in the middle dimension of \(A({\mathbb{C}})\) modulo torsion. The role of S is played for example by an appropriate power of the cohomology class of a hyperplane section (giving a one-dimensional S), in another example the role of S is played by the lattice S(A) spanned by the homology classes of the orientable connected components of the set \(A({\mathbb{R}})\) of real points and \(\theta\) is the identity. As a result the author gets new invariants of real algebraic surfaces and new relations among these invariants, he also gets new relations among standard invariants such as the Euler characteristic of the set of real points \(A({\mathbb{R}})\) and the Euler characteristics of the orientable connected components of \(A({\mathbb{R}})\). K3-surfaces; integral symmetric bilinear form; involution; system of invariants; genus; isomorphism classes; real nonsingular projective algebraic variety; new invariants of real algebraic surfaces; relations; standard invariants; Euler characteristic Nikulin V.V. (1984). Involutions of integral quadratic formsand their applications to real algebraic geometry. Math. USSR Izv. 22: 99--172 General binary quadratic forms, Quadratic forms over global rings and fields, Surfaces and higher-dimensional varieties, Real algebraic and real-analytic geometry Involutions of integral quadratic forms and their applications to real 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 The aim of the present article is to construct analytic invariants for a germ of a holomorphic function having a one-dimensional critical locus \(S\). This is done for a large class of such germs containing for instance any quasi-homogeneous germ at the origin. More precisely, aside from the Brieskorn \((a,b)\)-module at the origin and a (locally constant along \(S^* := S\{0\}\)) sheaf \({\mathcal H}^n\) of \((a, b)\)-modules associated with the transversal hypersurface singularities along each connected component of \(S^*\), we construct also \((a,b)\)-modules with supports \(E_c\) and \(E'_{c\cap S}\).
An interesting consequence of the local study along \(S^*\) is the corollary showing that for a germ with an isolated singularity, the largest sub-\((a, b)\)-module having a simple pole in its Brieskorn-\((a,b)\)-module is independent of the choice of a reduced equation for the corresponding hypersurface germ.
We also give precise relations between these various \((a,b)\)-modules via an exact commutative diagram. This is an \((a, b)\)-linear version of the tangling phenomenon for consecutive strata we have previously studied in the ``topological'' setting for the localized Gauss-Manin system of \(f\).
Finally, we show that in our situation there exists a non-degenerate \((a, b)\)-sesquilinear pairing
\[
h:E\times E_{e\cap S}'\to |\Xi'|^2
\]
where \(||\Xi'|^2\) is the space of formal asymptotic expansions at the origin for fiber integrals. This generalizes the canonical Hermitian form defined in 1985 for the isolated singularity case (for the \((a, b)\)-module version see the recent 2005 paper). Its topological analogue (for the eigenvalue 1 of the monodromy) is the non-degenerate sesquilinear pairing
\[
h: H^n_{c\cap S}(F,\mathbb C)_{=1}\times H^m(F,\mathbb C)_{=1}\to \mathbb C
\]
defined in an earlier paper for an arbitrary germ with a one-dimensional critical locus. Then, we show this sesquilinear pairing is related to the non-degenerate sesquilinear pairing introduced on the sheaf \({\mathcal H}^n\) via the canonical Hermitian form of the transversal hypersurface singularities.
For part I of this paper see [\textit{D. Bartlet}, Bull. Soc. Math. Fr. 134, No.~2, 173--200 (2006; Zbl 1126.32022).] Barlet, D, Sur certaines singularités d'hypersurfaces II, J. Algebraic Geom., 17, 199-254, (2008) Complex surface and hypersurface singularities, Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants, Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties On certain 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 This work introduces a new framework for understanding uniform behavior of singularity measures such as Hilbert-Kunz multiplicity, Hilbert-Samuel multiplicity, and F-rational signature, for ideals varying in families of rings. Namely, the author calls the combination of a ring map $R \rightarrow A$ with an ideal $I$ of $A$ an \textit{affine $I$-family} if $A/I$ is module-finite over $R$, $I \cap R = 0$, and certain dimension formulas hold. (This is a restricted version of Lipman's notion of an $I$-family from [\textit{J. Lipman}, Lect. Notes Pure Appl. Math. 68, 111--147 (1982; Zbl 0508.13013)]). Then one analyzes the ideals $I(\mathfrak p)$, $I(\mathfrak p)^{[p^e]}$ (when char $k(\mathfrak p) = p>0$) and $I(\mathfrak p)^n$ for $\mathfrak p \in $Spec$(R)$ and various $e, n \in \mathbb N$ in the rings $R(\mathfrak p)$, where the notation $(\mathfrak p)$ means to tensor over $R$ with the residue field $k(\mathfrak p)$ of $\mathfrak p$.
This framework allows the author to recover Lipman's result [loc. cit.] on upper semicontinuity of Hilbert-Samuel multiplicity on the prime spectrum, by showing that the \textit{terms} defining Hilbert-Samuel multiplicity as a limit are also upper semicontinuous in the family.
He also recovers some of his own results (see [Compos. Math. 152, No. 3, 477--488 (2016; Zbl 1370.13006)]) on semicontinuity of Hilbert-Kunz multiplicity on the prime spectrum, again by analyzing the terms, where in this case one has an affine $I$-family $R \rightarrow S$ with reduced fibers of dimension $=$ height$(I)$, where $R$ is F-finite. He further obtains upper semicontinuity of Hilbert-Kunz multiplicity in an affine family where char $R=0$ and the characteristics of the fibers can vary but all residue fields of $R$ are F-finite when they are positive characteristic. In particular, when $R = \mathbb Z$, this answers a question of Claudia Miller from [\textit{H. Brenner} et al., J. Algebra 372, 488--504 (2012; Zbl 1435.13014)] in pursuance of obtaining a sensible notion of Hilbert-Kunz multiplicity in equal characteristic zero.
The author also parlays his methods to show that for local algebras essentially of finite type over a prime characteristic field, the infimum in Hochster and Yao's definition of F-rational signature is actually achieved. He thus recovers a special case of the result in [\textit{M. Hochster} and \textit{Y. Yao}, ``F-rational signature and drops in the Hilbert-Kunz multiplicity'', Preprint] that the F-rational signature of the ring is positive if and only if the ring is F-rational. multiplicity; Hilbert-Samuel polynomial; Hilbert-Kunz multiplicity; semicontinuity; families Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Multiplicity theory and related topics, Singularities in algebraic geometry, Deformations of singularities, Fibrations, degenerations in algebraic geometry On semicontinuity of multiplicities in families | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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{B. R. Greene} and \textit{M. R. Plesser} [Nuclear Phys. B 338, No. 1, 15-37 (1990)] showed explicitly how to construct a mirror Calabi-Yau for a hypersurface in weighted projective space which is of Fermat type. This is done by factoring out by a subgroup of the (finite) symmetry group and resolving singularities. In this paper the authors give a very natural extension to certain hypersurfaces of non-Fermat type. They require only that the number of monomials be equal to the number of variables (five in the case of hypersurfaces of weighted projective space); they associate to such a polynomial a matrix \(P = (p_{ij})\),where \(p_{ij}=\) power of the \(i\)th variable in the \(j\)th monomial. As they show, the transposed matrix \(^TP\) determines again a Calabi-Yau hypersurface, and the finite group of symmetries \(G\) of the original hypersurface \(X\) again acts on this transpose \(^TX\). Then they find a subgroup \(H \subset G\) of this symmetry group, such that a candidate for the mirror \(X^m\) of \(X\) is given by \(^TX/HS\) (with singularities resolved). By candidate we mean that the authors have checked that the Hodge diamond are mirrors, but not that the Yukawa couplings coincide. Since for a Fermat polynomial the transpose coincides with \(P\), this construction reduces in this case to that of Greene and Plesser [loc. cit.].
These considerations are applied to give an example of a (candidate for a) mirror for one of the manifolds listed in a paper by \textit{P. Candelas, M. Lynker} and \textit{R. Schimmrigk} [Nuclear Phys. B 341, No. 2, 383-402 (1990)] which is listed without a mirror. This is also applied to describe a mirror for the ``\(D^k\), \(N = 2\) superconformal minimal model'' (where the superpotential is given by the polynomial of \(D_k\) type \(x^{k-1}_1 + x_1 x^2_2\)). construction of mirror manifolds; Calabi-Yau hypersurface; Fermat type; Hodge diamond P. Berglund and T. Hubsch, \textit{A generalized construction of Calabi-Yau models and mirror symmetry}, arXiv:1611.10300 [INSPIRE]. Compact complex \(3\)-folds, Applications of deformations of analytic structures to the sciences, \(3\)-folds A generalized construction of mirror manifolds | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper deals with the problem of characterizing quasihomogeneous isolated singularities. The history begins in 1971 with the beautiful result of \textit{K. Saito} [Invent. Math. 14, 123-142 (1971; Zbl 0224.32011)]: an isolated complex hypersurface singularity with defining equation \(f\) is quasihomogeneous (i.e., after a change of coordinates \(f\) can be made into a quasihomogeneous polynomial) if and only if \(f \in j(f)\), where \(j(f)\) is the ideal generated by the partial derivative of \(f\) (this ideal is also called the jacobian ideal of \(f)\). -- In the subsequent years this result was extended to other fields and significantly generalized.
In this paper we further explore the properties of quasihomogeneous isolated singularities via module-theoretic techniques. Our main tool is what we call the gluing construction for Cohen-Macaulay modules which produces minimal Cohen-Macaulay approximations (and, if the ring is Gorenstein, hulls of finite injective dimension) for (nonmaximal) Cohen- Macaulay modules and their syzygy modules. Gorenstein modules; complete intersections; quasihomogeneous isolated singulrities; hypersurface singularity; jacobian ideal; minimal Cohen- Macaulay approximations Herzog, Jürgen; Martsinkovsky, Alex: Gluing Cohen-Macaulay modules with applications to quasihomogeneous complete intersections with isolated singularities, Comment. math. Helv. 68, No. 3, 365-384 (1993) Singularities of surfaces or higher-dimensional varieties, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Complete intersections, Cohen-Macaulay modules, Local complex singularities Gluing Cohen-Macaulay modules with applications to quasihomogeneous complete intersections with 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 Stacks provide a unified conceptual framework for treating equivariant problems about \(G\)-spaces in geometry. In algebraic geometry, the theory of algebraic stacks was basically invented by Deligne and Mumford (1969), largely by further developing some earlier pioneering ideas of Grothendieck, and extended later on by M. Artin. Algebraic stacks have ever since become an indispensable tool in algebraic geometry, mainly in the study of moduli problems via suitable quotient constructions. In the more general context of topology and complex-analytic geometry, however, stacks have only appeared in special ad-hoc forms here and there, mostly under different names such as ``graphs of groups'' (Bass and Serre) and ``orbifolds'' (Thurston). It was only very recently that a general, coherent and solid theory of topological stacks, including also analytic stacks, has begun to be built up by B. Noohi, the second author of the paper under review [cf. \textit{B. Noohi}, Foundations of topological stacks I, \url{arXiv:math/0503247}]. In this general context, analytic Deligne-Mumford stacks appear as very natural objects, together with their own homotopy and (co)homology theories, quotient constructions, and other classically familiar concepts.
Especially analytic Deligne-Mumford stacks of dimension one, accordingly called ``(analytic) Deligne-Mumford curves'', appear to be of great interest, since these objects naturally generalize ordinary Riemann surfaces and their particular geometry.
In this vein, the paper under review is devoted to the related question of to what extent the classical uniformization theory of Riemann surfaces can be generalized to analytic Deligne-Mumford curves.
Based on B. Noohi's general framework of topological and analytic stacks, which is used as a crucial ingredient throughout the whole exposition, the authors provide a complete answer to this problem.
In the case of Riemann surfaces, the uniformization theorem states that there are three one-dimensional analytic, domains, namely \({\mathfrak H}\) (the Siegel upper-half plane), \(\mathbb{C}\) and \(\mathbb{P}^1(\mathbb{C})\), appearing as universal covering spaces of Riemann surfaces, and that every Riemann surface arises as a quotient of one of these simply connected domains by a discrete subgroup of its analytic automorphism group. These occurring discrete subgroups, being the fundamental groups of the respective Riemann surfaces, have a particularly simple form and can be completely described.
In passage to analytic Deligne-Mumford curves, the authors show that the situation is quite similar, however partly much more subtle. One of their main theorems establishes the fact that the simply connected Deligne-Mumford curves are precisely \({\mathfrak H}\), \(\mathbb{C}\) and the so-called ``weighted projective lines'' \({\mathcal P}(m,n)\) for arbitrary \((m,n)\in\mathbb{N}\times\mathbb{N}\), the latter ones being constructed as in B. Noohi's fundamental preprint cited above.
This implies that the familiar trichotomy of hyperbolic, Euclidean, and spherical Riemann surfaces continues to hold, mutatis mutandis, in the more general theory of Deligne-Mumford curves. Furthermore, the authors show how to compute the (Noohi) fundamental groups of non-singular analytic Deligne-Mumford curves, and how to read off their uniformization, type by topological inspection. Then, representing an analytic Deligne-Mumford curve as a quotient (stack) of its universal cover (\({\mathfrak H}\), \(\mathbb{C}\) or \({\mathcal P}(m,n)\), respectively), where the spherical case turns out to be much more delicate than the hyperbolic or Euclidean ones and requires some advanced 2-group theory, the authors obtain a complete classification of analytic Deligne-Mumford curves by their uniformization type as defined before. Methodologically, the uniformization problem for analytic Deligne-Mumford curves is first tackled in the orbifold case (à la Thurston) by using the Van Kampen theorem, and is then extended to the general case of smooth Deligne-Mumford curves by applying the theory of graphs of groups (à la Bass-Serre) in addition. Along the way, the authors prove a number of auxiliary, often very general results that are of independent interest, including the fact that every Deligne-Mumford stack can be interpreted as a gerbe over an orbifold.
The relevant facts on 2-groups and crossed-modules, which are extensively used throughout the paper, are gathered in an appendix at the end. Deligne-Mumford stacks; homotopy theory; fundamental groups; uniformization; groupoids; Riemann surfaces; orbifolds; trees Kai Behrend & Behrang Noohi, ``Uniformization of Deligne-Mumford curves'', J. Reine Angew. Math.599 (2006), p. 111-153 Generalizations (algebraic spaces, stacks), Homotopy theory and fundamental groups in algebraic geometry, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Families, moduli of curves (analytic), Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Groupoids, semigroupoids, semigroups, groups (viewed as categories) Uniformization of Deligne-Mumford 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 \textit{A. Braverman} and \textit{M. Finkelberg} recently proposed the geometric Satake correspondence for the affine Kac-Moody group \(G_{\text{aff}}\) [Duke Math. J. 152, No. 2, 175--206 (2010; Zbl 1200.14083)]. They conjecture that intersection cohomology sheaves on the Uhlenbeck compactification of the framed moduli space of \(G_{\text{cpt}}\)-instantons on \(\mathbb R^{4}/\mathbb Z_r\) correspond to weight spaces of representations of the Langlands dual group \(G_{\text{aff}}^\vee\) at level \(r\). When \(G = \text{SL}(l)\), the Uhlenbeck compactification is the quiver variety of type \(\mathfrak{sl}(r)_{\text{aff}}\), and their conjecture follows from the author's earlier result and I. Frenkel's level-rank duality. They further introduce a convolution diagram which conjecturally gives the tensor product multiplicity [Braverman A., Finkelberg M., Private communication, 2008]. In this paper, we develop the theory for the branching in quiver varieties and check this conjecture for \(G = \text{SL}(l)\). quiver variety; geometric Satake correspondence; affine Lie algebra; intersection cohomology Nakajima, H., \textit{quiver varieties and branching}, SIGMA Symmetry Integrability Geom. Methods Appl., 5, (2009) Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Kac-Moody groups Quiver varieties and branching | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities \textit{V. Ginzburg} [C. R. Acad. Sci., Paris, Sér. I 312, No. 12, 907--912 (1991; Zbl 0749.17009)] and \textit{H. Nakajima} [in Duke Math. J. 76, No. 2, 365--416 (1994; Zbl 0826.17026)] have given two different geometric constructions of quotients of the universal enveloping algebra of \(U(\mathfrak {sl}_n)\)and its irreducible finite-dimensional highest weight representations using the convolution product in the Borel-Moore homology of flag varieties and quiver varieties, respectively. The purpose of this paper is to explain the precise relationship between the two constructions. In particular, we show that while the two yield different quotients of the universal enveloping algebra, they produce the same representations and the natural bases which arise in both constructions are the same. We also examine how this relationship can be used to translate the crystal structure on irreducible components of quiver varieties, defined by Kashiwara and Saito, to a crystal structure on the varieties appearing in Ginzburg's construction, thus recovering results of Malkin. geometric representation theory; convolution product; Lie algebras; crystal bases; quiver varieties Universal enveloping (super)algebras, Representations of quivers and partially ordered sets, Grassmannians, Schubert varieties, flag manifolds, Universal enveloping algebras of Lie algebras On two geometric constructions of \(U(\mathfrak {sl}_n)\) and its representations | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper discusses homological mirror symmetry correspondence between contractions and degenerations of Calabi--Yau varieties. In homological mirror symmetry, degenerations lead to symplectomorphisms while contractions give rise to Fourier-Mukai functors. The author reviews in particular the dictionary between symplectic isomorphisms and Fourier-Mukai functors.
In the first two sections a quick but exhaustive introduction to homological mirror symmetry is given, followed by the description of the relation of symplectomorphisms and derived equivalences arising from mirror symmetry between degenerations and contractions of Calabi--Yau varieties, which is the main subject of this note. The second part of the paper is devoted to the description of \(2\) and \(3\) dimensional cases. In dimension \(2\), the only class of birational contractions of Calabi--Yau varieties is given by contractions of \((-2)\)-curves on a \(K3\) surface. In dimension \(3\), three different cases are possible depending on the dimensions of the exceptional locus and of its image. The author then provides known results and conjectural examples in all of these cases. homological mirror symmetry; Calabi-Yau threefolds; \(K3\) surfaces; contractions and degenerations Szendrői, Balázs, Contractions and monodromy in homological mirror symmetry.Strings and geometry, Clay Math. Proc. 3, 301-314, (2004), Amer. Math. Soc., Providence, RI Calabi-Yau manifolds (algebro-geometric aspects), Global theory and resolution of singularities (algebro-geometric aspects), Derived categories, triangulated categories Contractions and monodromy in homological mirror symmetry | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [This article was published in the book announced in Zbl 0342.00010].
From the author's introduction: ``The rational double points of surfaces in characteristic zero are related to the finite subgroups \(G\) of \(SL_2\). Namely, if \(V\) denotes the affine plane with its linear \(G\)-action, then the variety \(X=V/G\) has a singularity at the origin, which is the one corresponding to \(G\).
Let \(p\) be a prime integer. If \(p\) divides the order of \(G\), this subgroup will degenerate when reduced modulo \(p\), and the smooth
reduction of \(V\) will usually not be compatible with an equisingular reduction of \(X\). Nevertheless, it turns out that every rational double point in characteristic \(p\) has a finite (possibly ramified) covering by a smooth scheme. In this paper we prove the existence of such a covering by direct calculation, and we compute the local fundamental groups of the singularities.''
The paper has an excellent brief list of references from which the reader can
obtain a perspective of the subject. Bibliography Artin, M., Coverings of the rational double points in characteristic \textit{p}, (Complex Analysis and Algebraic Geometry, (1977), Iwanami Shoten Tokyo), 11-22, MR 0450263 Coverings in algebraic geometry, Rational points, Local ground fields in algebraic geometry, Singularities in algebraic geometry Coverings of the rational double points in characteristic p | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 discusses techniques relevant in the theory of resolution of singularities of algebraic varieties. It is largely an exposition of previous results of the authors and other researchers (Hironaka, Encinas, Benito, García-Escamilla, etc.), but also contains some new results. Let us explain its content more carefully.
Trying to resolve the singularities of an algebraic variety \(X\) over a field \(k\), an approach (due to Hironaka) is to associate to \(X\) an upper semicontinuous function \(F_X\) into a (fixed) totally ordered set \(\Lambda\), which in some sense measures how bad the singularities of \(X\) are.
The set \(C\) of points of \(X\) where \(F_X\) reaches its maximum value is closed. For suitable \(F_X\), blowing up \(C\) we should get a variety whose ``worst'' singularities are ``better '' than those of \(X\). In this paper, the cases where \(F_X=HS_X\) or \(F_X=e_X\), the Hilbert-Samuel or the multiplicity function respectively, are considered.
To control this process, an important concept is that of \textit{representation}. In general, an \textit{idealistic pair} (or just a pair) on a smooth variety \(W\) is an ordered pair \((J,b)\) where \(J\) is a coherent sheaf of ideals on \(W\) and \(b \geq 0\) an integer. The singular locus \(\mathrm{Sing}(J,b)\) of the pair is the set of points of \(W\) where \(\mu_x(J)\), the order of the stalk \(J_x\) in \({\mathcal O}_{W,x}\), is \(\geq b\). There is a natural notion of \textit{resolution of pairs} (eventually, the singular locus is empty). For an upper semicontinuous function \(F_X\) as above, if \(X\) is embedded as a closed subvariety of a regular \(W\), a \textit{representation of \(F_X\)} is a pair \((J,b)\) on \(W\) such that \(\mathrm{Sing}(J,b)=\mathrm{Max}(F_X)\). If this can be done in such a way that this equality is preserved after taking suitable blowing-ups (``permissible blowing-ups'') then from a resolution of the pair \((J,b)\) one reaches a situation where the maximum value of the function \(F\) has dropped.
For \(F_X=HS_X\) or \(F_X=e_X\) representation is available, locally in the étale topology. Pairs can be resolved in characteristic zero. In the usual proofs there is an essential inductive step (on the dimension of \(W\), in the previous notation), which uses the local existence of certain smooth subvarieties of \(W\), called of \textit{maximal contact}. These are not available over fields of positive characteristic, which prevents the argument to generalize. Villamayor proposed to replace these subvarieties of maximal contact by suitable projections onto smooth varieties of smaller dimension. To do induction successfully, it seems convenient not to use pairs as above, but essentially equivalent more algebraic objects, called \textit{Rees algebras}. These are certain graded subalgebras of \({\mathcal O}_W[T]\) (with \(T\) an indeterminate). To find the necessary suitable projections, a numerical invariant \(\tau\), also introduced by Hironaka, plays an important role. This provides an alternative presentation of the proof of resolution in characteristic zero. For instance, certain fundamental results on Rees algebras obtained by the authors allow them to greatly simplify the proof that the local process of representation mentioned before globalizes. But, more importantly, this approach might work also in positive characteristic. Some partial results were obtained by the authors.
All these developments, including many of the technicalities involved, as well as other topics, are discussed in the reviewed paper. A good part of the article is on the necessary commutative algebra. One of the new results presented is the theory of \textit{identifiable pairs}, useful when one must compare, in a suitable way, Rees algebras defined over different ambient spaces.
Several relevant examples are examined. Overall this article is a good introduction to the subject. resolution of singularities; blowing up; multiplicity; integral closure; elimination theory; Rees algebra; equivalence invariant Bravo, A., Villamayor, O.E.: On the behavior of the multiplicity on schemes: stratification and blow ups. In: Ellwood, D., Hauser, H., Mori, S., Schicho, J. (eds.) The Resolution of Singular Algebraic Varieties. Clay Institute Mathematics Proceedings, vol. 20, pp. 81-207 (340pp.). AMS/CMI, Providence (2014) Global theory and resolution of singularities (algebro-geometric aspects), Local structure of morphisms in algebraic geometry: étale, flat, etc., Singularities of surfaces or higher-dimensional varieties, Birational geometry, Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics, Integral closure of commutative rings and ideals, Multiplicity theory and related topics On the behavior of the multiplicity on schemes: stratification and blow ups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper extends the equivariant Gromov-Witten/stable pairs correspondence of Maulik-Oblomkov-Okounkov-Pandharipande from primary invariants to full descendent theory for non-singular quasi-projective toric \(3\)-folds. The correspondence is expressed as equality of renormalized partition functions under the change of variables \(-q=e^{iu}\), and a transformation of the descendent insertions. The transformation is given by a universal correspondence matrix \(K_{\alpha,\widehat{\alpha}}\) indexed by pairs of partitions and with values in \(\mathbb{Q}[i,w_1,w_2,w_3]((u))\). The matrix is triangular in the sense that its coefficients vanish for \(|\alpha|<|\widehat{\alpha}|\), and symmetric in the variables \(w_i\). The change of variables is well defined due to the rationality in \(q\) of the partition function of the stable pairs of toric \(3\)-folds. A parallel rationality result is expected to be false in the non-equivariant case, but the authors manage to rearrange \(K\) into a form which makes the existence of the non-equivariant limit explicit. Most applications of the correspondence are based on this non-equivariant limit.
As the authors note, the main obstacle to applications is that the \(u\) dependence of \(K\) ``remains mysterious'', but they are able to determine the leading terms, in particular that \(K_{\alpha,\alpha}=(iu)^{l(\alpha)-|\alpha|}\). This is enough to prove an explicit non-equivariant correspondence for invariants of projective toric \(3\)-folds with primary and stationary descendent insertions only. This validates a conjecture of Maulik-Nekrasov-Okounkov-Pandharipande in the toric case, and in turn allows to derive the rationality result for the Gromov-Witten partition function with such insertions from the stable pairs rationality. So far this result is inaccessible by other means. For primary fields only a relative version of the correspondence is proved for some log Calabi-Yau geometries, in particular nonsingular projective Fano toric \(3\)-folds with a nonsingular anti-canonical \(K3\) surface as the relative divisor \(D\). A rationality constraint on the non-equivariant descendent series of \(\mathbb{P}^3\) then follows. The only equivariant correspondence proved is for \((\mathcal{A}_n\times\mathbb{P}^1)/D\), where \(\mathcal{A}_n\) is the minimal toric resolution of the standard \(A_n\)-singularity, and \(D\) is the union of fibers over three distinct points, which resolves the issues left open in the work of Maulik-Oblomkov.
The construction of \(K\) is based on the \(1\)-leg equivariant descendent invariants, and the primary correspondence and the capped descendent vertex of the authors' earlier paper are used to extend the correspondence to the \(2\)-leg and \(3\)-leg cases. The divisibility properties of the coefficients of \(K\) needed to establish the existence of the non-equivariant limit are proved geometrically. projective toric 3-folds; partition function; capped descendent vertex; rationality constraint; projective Fano toric 3-folds; relative divisor; \(K3\) surface Bryan, J., Oberdieck, G., Pandharipande, R., Yin, Q.: Curve counting on abelian surfaces and threefolds. Algebr. Geom. arXiv:1506.00841\textbf{(to appear)} Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Vector bundles on curves and their moduli Gromov-Witten/pairs descendent correspondence for toric 3-folds | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper under review continues the study of the authors about singular quiver Grassmannians, providing desingularizations of irreducible components of arbitrary quiver Grassmannians over Dynkin quivers.
Given a Dynkin quiver \(\mathcal Q\), being a directed graph whose unoriented underlying graph is a Dynkin diagram, a representation \(M\) of \(\mathcal Q\), and a dimension vector \(e\), the quiver Grassmannian \(Gr_e(M)\) is the variety of subrepresentations of \(M\) of dimension vector \(e\). When \(M\) has good homological properties, \(Gr_e(M)\) is smooth. However, the general analysis needs a desingularization of the quiver Grassmannian.
The construction of a desingularization lies on the definition of an algebra \(B_{\mathcal Q}\) for the quiver \(\mathcal Q\) which has global dimension at most two, such that the original module category \(\mathrm{mod\,}k\mathcal Q\) embeds into the subcategory \(\mathrm{mod\,}B_{\mathcal Q}\) of objects of projective and injective dimensions at most one, where all non-trivial extensions in \(\mathrm{mod\,}k\mathcal Q\) vanish after the embedding. This is to avoid the natural embedding \(M\mapsto\Hom(-,M)\) of \(\mathrm{mod\,}k\mathcal Q\) into \((\mathrm{mod\,}k\mathcal Q)^{op}\) which give projective functors of dimension two, in general.
A quiver \(\widehat{\mathcal Q}\) is constructed from \(B_{\mathcal Q}\) and, for every representation \(M\) over \(\mathcal Q\), a representation \(\widehat M\) over \(\widehat{\mathcal Q}\) arises. This, together with a fully faithful functor \(\Lambda\) from the category of representations of \(\mathcal Q\) to the one in \(B_{\mathcal Q}\) with good homological properties, completes the ingredients of the construction.
Then, it is constructed the desingularization map \(\pi_{[N]}\colon Gr_{\dim\widehat N}(\widehat M)\to Gr_e(M)\), where \(\widehat N\) is a representation over \(\widehat{\mathcal Q}\) coming from an isomorphism class of representations \([N]\) over \(\mathcal Q\) of dimension vector \(e\). The variety \(Gr_{\dim\widehat N}(\widehat M)\) is smooth with irreducible equidimensional connected components, and the fibers of the map can be described in terms of a quiver Grassmannian over \(\widehat{\mathcal Q}\) itself, once we know the irreducible components of the singular \(Gr_e(M)\). The article finishes by showing some particular examples of the construction. quiver Grassmannians; desingularizations; Dynkin quivers; Auslander algebras; flag varieties; Auslander-Reiten theory; quiver representations; categories of representations; irreducible components Cerulli Irelli, G., Feigin, E., Reineke, M.: Desingularization of quiver Grassmannians for Dynkin quivers. Adv. Math. \textbf{245}, 182-207 (2013) Representations of quivers and partially ordered sets, Grassmannians, Schubert varieties, flag manifolds, Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers, Singularities of surfaces or higher-dimensional varieties Desingularization of quiver Grassmannians for Dynkin quivers. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(F\) be a field of characteristic zero, \(G\) a quasi-simple simply connected algebraic group over \(F\) and \(G(F)\) the group of \(F\)-rational points of \(G\). When \(G\) is \(F\)-isotropic one can consider a normal subgroup \(G(F)^+\) of \(G(F)\) generated by \(F\)-unipotent elements. It is known that \(G(F)^+\) is projectively simple as an abstract group and it was conjectured by M. Kneser and J. Tits that in fact \(G(F)=G(F)^+\). It is known that this conjecture is false in general and this implies that at least from the point of view of description of the normal structure of \(G(F)\) the groups \(W(G(F))=G(F)/G(F)^+\) are of great importance. As it was shown by Tits the following types of groups \(G\) (where \(W(G(F))\) can be non-trivial) are of main interest.
1. A special case. Let \(D\) be a finite-dimensional non-commutative central division algebra over \(F\), \(\text{GL}_n(D)\) (\(n>1\)) general linear group of \(D\) of degree \(n\) and \(\text{Nrd}_{\text{GL}_n}(D)\) the reduced norm homomorphism. Then \(G(F)=\text{Ker(Nrd}_{\text{GL}_n}(D))\).
2. A special unitary case. Let \(K/F\) be a separable quadratic field extension, \(D\) a finite-dimensional non-commutative central division algebra over \(K\) with an involution \(\tau\) of the second kind such that the subfield of invariants of \(\tau\) in \(K\) is \(F\). Let \(V_n\) (\(n>1\)) be a right \(n\)-dimensinal vector space over \(D\) and \(\Phi_n\colon V_n\times V_n\to D\) a non-degenerated skew-Hermitian form with respect to \(\tau\). For unitary group \(U(\Phi_n)\) of \(\Phi_n\) let \(G(F)=\text{SL}_n(D)\cap U(\Phi_n)\).
3. A spinor case. Let \(D/F\) be a non-commutative central division algebra of dimension \(m^2\) with an involution \(\tau\) of the first kind (i.e. \(f^\tau=f\) for any \(f\in F\)), \(S_\tau(D)=\{a\in D\mid a^\tau=a\}\). An involution \(\tau\) is called symplectic if \(\dim_FS_\tau(D)=m(m-1)/2\) and orthogonal if \(\dim_FS_\tau(D)=m(m+1)/2\).
For non-degenerated \(\tau\)-Hermitian form (\(\tau\) is orthogonal) on the right \(n\)-dimensional \(D\)-vector space \(\Phi_n\) let \(G(F)\) be the group of \(F\)-rational points of spinor group of \(\Phi_n\), we denote it by \(\text{Spin}(\Phi_n)\).
We refer to cases 1-3 as to the ones of algebraic groups of non-commutative classical types.
The aim of the paper is to generalize results on triviality of Whitehead groups in the above cases 1-3 to the class of fields of virtual cohomological dimension at most 2.
Whitehead groups \(W(G(F))\) are closely related to groups of \(R\)-equivalence classes of the corresponding varieties. At the end of the paper we also discuss these relations. simply connected almost simple groups; groups of rational points; unipotent radicals; parabolic subgroups; Whitehead groups; classical groups Vyacheslav I. Yanchevskiĭ, Whitehead groups and groups of \?-equivalence classes of linear algebraic groups of non-commutative classical type over some virtual fields, Algebraic groups and arithmetic, Tata Inst. Fund. Res., Mumbai, 2004, pp. 491 -- 505. Linear algebraic groups over arbitrary fields, \(K\)-theory of quadratic and Hermitian forms, Rational points Whitehead groups and groups of \(R\)-equivalence classes of linear algebraic groups of non-commutative classical type over some virtual fields. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities A new proof of Hironaka's theorem on resolution of singularities is given. There are already several different constructive approaches [cf. \textit{E. Bierstone} and \textit{P. Milman}, Invent. Math. 128, 207--302 (1997; Zbl 0896.14006) or \textit{O. Villamayor}, Ann. Sci. Éc. Norm. Supér., IV. Sér 22, 1--32 (1989; Zbl 0675.14003)]. The resolution process is based on the choice of an invariant which measures the singularities and drops under blowing up the maximal stratum of this invariant. The choice of the invariant and the way to compute it makes the difference between the approaches to resolve singularities. The definition of the invariant is quite involved. It is defined inductively using the knowledge of the resolution process up to this moment. The induction defining the invariant is given by the intersection of the variety with a so-called hypersurface of maximal contact. The choice of this hypersurface is not canonical and some effort is needed to define the invariant in a canonical way.
The approach of this paper is similar to the approach of Bierstone and Milman but based additionally on two observations. The resolution process defined as a sequence of blowing-ups the ambient spaces can be applied simultaneously to a class of equivalent singularities obtained by simple modifications, i.e. to resolve a singularity it is allowed to tune it before starting: In the equivalence class a convenient representative given by a so-called homogenized ideal is chosen. The restrictions of homogenized ideals to different hypersurfaces of maximal contact define locally analytically isomorphic singularities. resolution of singularities; algorithmic resolution Włodarczyk, Jarosław, Simple Hironaka resolution in characteristic zero, J. Amer. Math. Soc., 18, 4, 779-822, (2005) Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Local complex singularities, Modifications; resolution of singularities (complex-analytic aspects) Simple Hironaka resolution in characteristic zero | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 Higgs bundle on a complex manifold \(X\) is a pair \((X,\theta)\), where \(E\) is a holomorphic vector bundle on \(X\) and and \(\theta\in H^0(X,\mathcal End(E)\otimes\Omega^1_X)\), such that \(\theta\wedge\theta=0\). Hitchin pairs \((E,L,\theta)\) are generalizations of Higgs bundles: \(L\) is a holomorphic bundle on \(X\) and now \(\theta\in H^0(X,\mathcal End(E)\otimes L).\) This text study infinitesimal deformations of Higgs bundles and Hitchin pairs and give a description of the Hitchin map as a morphism of deformation theories; differential graded Lie algebras are used to analyze these deformations.
The philosophy through the article is that in characteristic zero, every deformation problem is governed by a differential graded algebra, via the deformation functor associated to it, given by solutions of Maurer-Cartan equation modulo gauge action. Dglas techniques allow to preserve a lot of information in the deformation problem, and some classical results on first order deformations and obstructions can be obtained as easy consequences of definitions and formal constructions.
One of the article's main goals is to find dglas that govern infinitesimal deformations of a Higgs bundle, of a pair (manifold, Higgs bundle), and of a Hitchin pair, and to obtain a description of first order deformations and obstructions from them. For the general case of deformations of a pair (manifold, Higgs bundle)\(=(X,E,\theta)\) the meaningful complex is
\[
\mathcal K:0\rightarrow\mathcal D^1(E)\overset{[-,\theta]}\rightarrow\mathcal End(E)\otimes\Omega^{1}_{X}\overset{[-,\theta]}\rightarrow\mathcal End(E)\otimes\Omega^{2}_{X}\overset{[-,\theta]}\rightarrow\mathcal End(E)\otimes\Omega^{3}_{X}\rightarrow\cdots,
\]
where the differential \([-,\theta]\) at the level of \(\mathcal End(E)\otimes\Omega^\ast_X\) is given by the composition of endomorphisms and the wedge product of forms, while at the first level it includes also the action of differential operators on forms via the Lie derivative.
A dgla is obtained, governing the infinitesimal deformations of the pair \((X,E,\theta)\), defining a dgla structure on the total complex of the Dolbeault resolutions of the above complex of sheaves
\[
\oplus_{p+q=\ast,p<\ast}A^{0,p}_X(\mathcal End(E)\otimes\Omega^q_X)\oplus A^{0,\ast}_X(\mathcal D^1(E)).
\]
So the first main result is stated as:
\textit{The space of first order deformations of \((X,E,\theta)\) is canonically isomorphic to the first hypercohomology space of the complex of sheaves \(\mathcal K\) and obstructions are contained in the second hypercohomology space of it.}
For the particular case of deformations of a Higgs bundle \((E,\theta)\), \(\mathcal K\) is substituted with the complex of sheaves:
\[
0\rightarrow\mathcal End(E)\overset{[-,\theta]}\rightarrow\mathcal End(E)\otimes\Omega^1_X\overset{[-,\theta]}\rightarrow\mathcal End(E)\otimes\Omega^2_X\overset{[-,\theta]}\rightarrow\mathcal End(E)\otimes\Omega^3_X\rightarrow\cdots
\]
for the general case of a Hitchin pair \((E,L,\theta)\), the complex to be considered is obtained by substituting in the above complex the sheaf \(\Omega^\ast_X\) with \(\bigwedge^\ast L\).
The second main goal of the article is the study of the Hitchin map from deformations point of view. It is defined as
\[
H:\mathcal M\rightarrow\oplus_{k=1}^r H^0(X,\odot^k L),\text{ }H(E,\theta)=(\text{Tr}(\theta),\dots,\text{Tr}(\theta^r)),
\]
from the moduli space \(\mathcal M\) of Hitchin pairs on \(X\) to the space of global sections of the vector bundles \(\odot^k L\). The Hitchin map was induced by Simpson as a generalization of the determinant map studied by Hitchin in the curve case.
To study the Hitchin map in terms of deformation theory, the dgla approach is not convenient, and the author use the tool of \(L_\infty\) algebras. Theory of deformations via differential graded Lie algebras and via \(L_\infty\)-algebras is based on the principle that the local study of a moduli space is encoded by a dgla or an \(L_\infty\)-algebra conveniently chosen. Then every natural morphism between moduli spaces is induced by a morphism between the associated dglas or \(L_\infty\)-algebras. The authors make explicit an \(L_\infty\)-morphism \(h\) that induces the Hitchin map. As a direct consequence of this \(L_\infty\)-description and of \(L_\infty\)-techniques, the following result is obtained:
\textit{The obstructions to deform Hitchin pairs are contained in the kernel of the map induced at second cohomology level by the linear part of \(h\).}
The article goes through the deformation theory both in the dgla and the \(L_\infty\) setting. The definitions and calculations are explicitly given, and is of great value. The results given are interesting for studying the corresponding moduli spaces, and the treatment and definition of the Hitchin map as induced by a map on \(L_\infty\)-level is very interesting. This principle could be used for the study of corresponding maps. All in all a very nice treatment and introduction to deformation theory and also to \(L_\infty\)-algebras. Hitchin pairs; dgla; differential graded Lie algebras; \(L_\infty\) algebras; governing; obstructions; hyper cohomology doi:10.1142/S0129167X1250053X Homological methods in Lie (super)algebras, Formal methods and deformations in algebraic geometry, Algebraic moduli problems, moduli of vector bundles Infinitesimal deformations of Hitchin pairs and Hitchin map | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities A singularity is said to be exceptional, if for any log canonical boundary, there is at most one exceptional divisor of discrepancy \(-1\). This notion is important for the inductive treatment of the log canonical singularities. -- The exceptional singularities of dimension 2 are known: They belong to the types \(E_6\), \(E_7\), \(E_8\) after Brieskorn. In this paper it was proved that the quotient singularity defined by Klein's simple group in its 3-dimensional representations is exceptional.
In the present paper, the classification of all the 3-dimensional exceptional quotient singularities is obtained. The main lemma states that the quotient of the affine 3-space by a finite group is exceptional if and only if the group has no semi-invariants of degree 3 or less. It is also proved that for any positive \(\varepsilon\), there are only finitely many \(\varepsilon\)-log terminal exceptional 3-dimensional quotient singularities.
Theorem: A 2-dimensional quotient singularity \(X=\mathbb{C}^2/G\) by a finite group \(G\) without reflections is exceptional if and only if \(G\) has no semi-invariants of degree \(\leq 2\).
Theorem: A 3-dimensional quotient singularity \(X=\mathbb{C}^3/G\) by a finite group \(G\) without reflections is exceptional if and only if \(G\) has no semi-invariants of degree \(\leq 3\). log canonical singularity; exceptional singularities; exceptional 3-dimensional quotient singularities Yu. G. Prokhorov and D. Markushevich, ''Exceptional Quotient Singularities,'' Am. J. Math. 121, 1179--1189 (1999). Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Singularities of curves, local rings, Homogeneous spaces and generalizations 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 The author classifies simple hypersurface K3 singularities (X,x) defined by a non-degenerated polynomial \(f(z_ 0,z_ 1,z_ 2,z_ 3)\) and he studies the minimal resolution \(\pi: (\tilde X,E)\to (X,x)\) and the singularities on the exceptional divisor E.
A three-dimensional singularity (X,x) is a simple K3 singularity if (X,x) is quasi-Gorenstein and if the exceptional divisor E of any minimal resolution is a normal K3 surface, where a minimal resolution \(\pi: (\tilde X,E)\to (X,x)\) is a proper morphism with only terminal singularities on \(\tilde X,\) with \(\tilde X\simeq X\setminus \{x\}\) and with \(K_{\tilde X}\) nef with respect to \(\pi\).
- The simple K3 singularities could be regarded as three-dimensional generalizations of simple elliptic singularities.
If the simple K3 singularity (X,x) is defined by a non-degenerated polynomial f(z), then \((1,1,1,1)\in \Gamma(f)\). The weight \(\alpha =\alpha(f)=(\alpha_ 1,\alpha_ 2,\alpha_ 3,\alpha_ 4)\) of the quasi-homogeneous polynomial \(f_{\Delta_ 0}\) associated to the face \(\Delta_ 0\) containing (1,1,1,1) verifies \(\sum^{4}_{i=1}\alpha_ i =1\). - Then to classify the simple K3 singularities we need to study the set \(W_ 4\) of weights: \(W_ 4=\{\alpha \in {\mathbb{Q}}^ 4_+| \quad \sum^{4}_{i=1}\alpha_ i =1,\alpha_ 1\geq...\geq \alpha_ 4\) and \((1,1,1,1)\in Int(C(\alpha))\},\) where \(C(\alpha\)) is the closed cone in \({\mathbb{R}}^ 4\) generated by the set \(T(\alpha)=\{\nu \in {\mathbb{Z}}^ 4_ 0| \alpha.\nu =1\}.\)
The author shows that the cardinality of \(W_ 4\) is 95, and for each weight \(\alpha\) he gives a quasi-homogeneous f of weight \(\alpha\) which defines a simple K3 singularity and such that \(\Delta_ 0=\Gamma (f)\) is the convex hull of \(T(\alpha\)). Then he constructs a minimal resolution \(\pi: \tilde X\to X\) using torus embedding: if the weight \(\alpha(f)=(p_ 1/p,...,p_ 4/p)\), where \(p_ 1,...,p_ 4\) are relatively prime integers, the filtered blow-up with weight \((p_ 1,...,p_ 4)\), \(\Pi: (V,F)\to ({\mathbb{C}}^ 4,0)\), induces a minimal resolution of (X,x). In the last part the author shows that under some conditions on the nondegenerate polynomial f defining the simple K3 singularity (X,x), the type and the number of the singularities on E are determined by the weight \(\alpha(f)\), independently of f. type of singularities; simple hypersurface K3 singularities; minimal resolution; exceptional divisor; number of the singularities; weight Yonemura, T., Hypersurface simple \textit{K}3 singularities, Tohoku Math. J. (2), 42, 3, 351-380, (1990), MR 1066667 Singularities of surfaces or higher-dimensional varieties, \(K3\) surfaces and Enriques surfaces, Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry Hypersurface simple K3 singularities | 0 |
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