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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 In the article under review, the authors study two invariants \(f^{(1,1)}\) and \(g^{(1,1)}\) for rational surface singularities. If \((M,A)\to (V,0)\) is a resolution of a normal surface singularity \((V,0)\) with exceptional set \(A\), then \(f^{(1,1)}=\dim \Gamma(\Omega_M^2)/\langle \Gamma(\Omega_M^1) \wedge \Gamma(\Omega_M^1) \rangle\) and \(g^{(1,1)}\) is defined in a similar way on \(M\setminus A\). Yau conjectured that these invariants are strictly positive for all normal surface singularities, and \textit{R. Du} and \textit{S. Yau} [Commun. Anal. Geom. 18, No. 2, 365--374 (2010; Zbl 1216.32018); J. Differ. Geom. 90, No. 2, 251--266 (2012; Zbl 1254.32051)] confirmed this conjecture for weighted homogeneous singularities, rational double points, and cyclic quotient singularities. In this paper, the authors prove that \(f^{(1,1)}= g^{(1,1)}\geq 1\). As an application, they solve the regularity problem of the Harvey-Lawson solution to the complex Plateau problem for a strongly pseudoconvex compact rational CR manifold of dimension three. The second half of the article devoted to the explicit calculation to show \(f^{(1,1)}= g^{(1,1)}= 1\) for rational triple points using local coordinates on the minimal resolution space. rational surface singularity; rational triple points; complex Plateau problem; strongly pseudoconvex; CR manifold Shokurov, V.V.: Problems about Fano varieties. In: Birational Geometry of Algebraic Varieties. Open Problems-Katata, pp. 30-32 (1988) Singularities in algebraic geometry, Complex surface and hypersurface singularities, CR manifolds as boundaries of domains New invariants for complex manifolds and rational singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities It is well-known that the moduli space \(M_g\) of compact Riemann surfaces of genus \(g\) is an irreducible quasiprojective subvariety of the moduli space \({\mathcal A}_g\) of principally polarized Abelian varieties of dimension \(g\), whose closure \(\overline M_g\) -- in the four-dimensional case -- turns out to be the exact zero-set of a theta-function. This theta-function is in fact a cusp form of weight 8 of the Siegel modular group, the so-called Schottky modular form. In the present paper, this fine result is achieved quite naturally by thorough geometric considerations involving the Gauss map on the theta divisor.
To be more precise, for a \(g\)-dimensional principally polarized Abelian variety \((A,\beta)\) with marked odd theta characteristic \(\xi\), the Taylor expansion about the origin of the theta function
\[
\vartheta[\xi](z, \Omega)= l(z, \Omega)+ m(z, \Omega)+\cdots
\]
yields a linear form \(l\) and a cubic form \(m\) in \(z\) such that all the ideals \((l),(l, m),\dots\subset \mathbb{C}[z]\) transform by the same automorphy factor under the action of a suitable subgroup of finite index of the Siegel modular group \(\text{Sp}(2g, \mathbb{Z})\). Especially, if \((A, \theta)\) is the Jacobian of a generic curve of genus \(g\), then for any choice of odd theta characteristics the restriction of \(\overline m\) to the hyperplane \(l= 0\) is a Fermat cubic, i.e. the sum of \(g- 1\) cubes.
In a second step, to any homogeneous polynomial invariant \(\varphi\) on cubic forms a natural globalization \(G_\varphi\) is constructed in such a way that the Siegel modular form \(G_\varphi(\overline m)\) vanishes on Jacobians if the invariant \(\varphi\) vanishes on the Fermat cubic.
Restricting to dimension form, the Fermat ideal generated by all Siegel modular forms coming from those invariants which vanish on the Fermat cubic is principal. Its zero set is exactly the closure of the locus of period matrices of genus four Riemann surfaces and a generator of this ideal equals, up to a constant multiple, the Schottky modular form. Siegel modular forms; cubic hypersurfaces; zero set of the Fermat ideal; Schottky modular form; Gauss map on the theta divisor; principally polarized Abelian variety; theta function; Siegel modular group; Jacobian of a generic curve; Fermat cubic; cubic forms Mccrory, C.; Shifrin, T.; Varley, R.: Siegel modular forms generated by invariants of cubic hypersurfaces. J. algebraic geom. 4, 527-556 (1995) Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms, Theta functions and curves; Schottky problem, Modular and Shimura varieties, Abelian varieties of dimension \(> 1\) Siegel modular forms generated by invariants of cubic hypersurfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities It was shown by \textit{A. Andreotti} and \textit{A. L. Mayer} [Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 21, 189--238 (1967; Zbl 0222.14024)] that the principally polarized abelian varieties \((A,\Theta)\) with a singular theta divisor form a divisor \(N_0\) in the moduli space \({\mathcal A}_g\) of ppav's. Mumford and Debarre showed that \(N_0\) has two irreducible components \(\theta_{\text{null}}\) and \(N_0'\), where \(\theta_{\text{null}}\) is the locus of ppav's whose theta divisor has a singularity at a point of order two, and \(N_0\) is the closure of the locus of those ppav's for which the theta divisor has a singularity at a point not of order two.
In this paper the locus \(\theta^{g-1}_{\text{null}}\subset \theta_{\text{null}}\) where the singularity is not an ordinary double point is studied. First it is shown that \(\theta^{g-1}_{\text{null}}\subsetneqq \theta_{\text{null}}\) (a different proof of this fact was given by Debarre). Then it is shown that \(\theta^{g-1}_{\text{null}}\) is contained in the intersection \(\theta_{\text{null}}\cap N_0'\). Moreover, using the geometry of the universal scheme of singularities of the theta divisor, it is worked out which components of this intersection are in \(\theta^{g-1}_{\text{null}}\). The proofs use theta function- and degeneration methods. Grushevsky, S., Salvati Manni, R.: Singularities of the theta divisor at points of order two. Int. Math. Res. Not. (2007). 10.1093/imrn/rnm045 Theta functions and curves; Schottky problem, Algebraic theory of abelian varieties, Theta functions and abelian varieties Singularities of the theta divisor at points of order two | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We present our recent understanding on resolutions of Gorenstein orbifolds, which involves the finite group representation theory. We concern only the quotient singularity of hypersurface type. The abelian group \(A_r(n)\) for \(A\)-type hypersurface quotient singularity of dimension \(n\) is introduced. For \(n=4\), the structure of the Hilbert scheme of group orbits and crepant resolutions of \(A_r(4)\)-singularity are obtained. The flop procedure of 4-folds is explicitly constructed through the process. crepant resolutions; Gorenstein quotient singularities; McKay correspondence Global theory and resolution of singularities (algebro-geometric aspects), Parametrization (Chow and Hilbert schemes), Singularities of surfaces or higher-dimensional varieties Orbifolds and finite group 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 contains two fundamental contributions.
{1. Dualizability.} The author studies dualizability for commutative group stacks.
{2. Albanese morphism.} The author defines and generalizes scheme theoretic Albanese torsors and morphisms to stacks.
The author then uses these fundamental contributions to describe universal torsors of Colliot-Thélène and Sansu in terms of the Albanese torsor. The author also uses these fundamental results to prove that Grothendieck's section conjecture holds for some root stacks over Severi-Brauer varieties.
Let us now describe the two main contributions of the paper.
{1. Dualizability.}
Let \(G\) be a commutative group stack \(G\) over a base scheme \(S\). One associated to \(G\) a commutative group stack \(D(G)\) defined as \(\mathscr{H}om (G , B \mathbb{G}_m) \). There is a canonical map \(e_G : G \to D (D(G))\). Naturally, one says that \(G\) is dualizable if \(e_G\) is an isomorphism. After recalling that Deligne \(1\)-motives and Beilinson \(1\)-motives are dualizable, the author attacks the question of finding sufficient conditions for \(G\) to be dualizable. The author has the following Conjecture.
{Conjecture.} Let \(S\) be a scheme with \(2 \in \mathcal{O} _S ^{\times}\). Let \(G\) be a proper, flat, and finitelly generated algebraic commutative group stack over \(S\), with finite and flat inertia stack. Then:
(1) \(D(G)\) is algebraic, proper, flat and finitely presented, with finite diagonal.
(2) \(G\) is dualizable.
The author has fundamental results in this direction. Under the assumptions of the conjecture, Brochard proves that \(D(G)\) is algebraic and finitely presented with quasi-finite diagonal and proves that (2) is a consequence of (1). Moreover the author proves the conjecture with the additional assumption that \(H^0(G)\) is cohomologically flat. The author gives other results with the same flavour in the paper. The author notes that the assumption that \(2 \in \mathcal{O} _S ^{\times}\) might be superfluous.
{2. Albanese morphism.}
Let \(X\) be an algebraic stack over a base \(S\) and denote by \(f: X \to S\) its structural morphism. Assume that \(\mathcal{O} _X \to f_* \mathcal{O} _X\) is universally an isomorphism and that \(f\) locally has sections in the \(fppf\) topology. The author defines the Albanese stack \(A^0_{\tau}(X)\) as the dual \(D(\mathrm{Pic}^{\tau} _{X/S})\) of the torsion component of the the Picard stack. Then Brochard defines an \(A_{\tau}^0(X)\)-torsor \(A^1_{\tau}(X)\) and a morphism \(a_X : X \to A_{\tau}^1(X)\) called the Albanese morphism. After introducing a ''duabelian condition'', the author proves the following Theorem.
{Theorem.} Assume that \(\mathrm{Pic}^{\tau} _{X/S}\) is a duabelian group. Then the Albanese morphism
\[
a_X : X \to A^1 _{\tau} (X)
\]
is initial among maps to torsors under abelian stacks, in the following sense. For any triple \((B, T, b)\) where \(B\) is an abelian stack, \(T\) is a \(B\)-torsor and \(b : X \to T\) is a morphism of algebraic stacks, there is a triple \((c_0, c_1, \gamma)\) where \(c_0 : A^0 _{\tau} (X) \to B\) is a homomorphism of commutative group stacks, \(c_1 : A^1 _{\tau} (X) \to T \) is a \(c_0\)-equivariant morphism, and \(\gamma\) is a 2-isomorphism \(c_1 \circ a_X \Rightarrow b.\) Such a triple \((c_0, c_1, \gamma)\) is unique up to a unique isomorphism. commutative groups stacks; dualizability; Picard stacks; Albanese morphism Generalizations (algebraic spaces, stacks), Picard schemes, higher Jacobians Duality for commutative group stacks | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities For \(\mathbf{G}\) an affine smooth group scheme over an Artinian local ring \(A\) with residue field \(k\) algebraically closed, one can associate to it a linear algebraic group \(G\) over \(k\) such that \(G\) is isomorphic to the \(A\)-valued points of \(\mathbf{G.}\) Under this association, \(G\) is the \(k\)-rational points of the group scheme \(\left( \mathcal{\mathfrak{F}}\mathbf{G}\right) \) given by \(\left( \mathcal{\mathfrak{F}}\mathbf{G}\right) \left( R\right) =\mathbf{G}\left( A\otimes_{W_{m}\left( k\right) }W_{m}\left( R\right) \right) ,\) where \(W_{m}\) is the scheme of truncated Witt vectors, and \(m=0\) if char \(A=0;\) otherwise, char \(A\) is a power of some prime \(p\) and we set\(\;\;m=\left( \log_{p}\text{char}A\right) -1.\) This functor coincides with what is usually called the Greenberg functor.
The main result of this paper establishes a connection between certain subgroup schemes of \(\mathbf{G}\) and certain subgroups of \(G\). Suppose that \(\mathbf{G}\) is a reductive group scheme, and that \(\mathbf{T}\) is a maximal torus of \(\mathbf{G.}\) Then the above functor maps \(\mathbf{T}\) to a Cartan subgroup of \(G\); furthermore this is a bijective correspondence between maximal tori and Cartan subgroups. As a consequence, all Cartan subgroups are conjugate in \(G\), providing some insight on the groups \(G\) that arise via the Greenberg functor. As a consequence of the proof of this result, it is shown that the normalizer respects the Greenberg functor, explicitly \(N_{\mathbf{G} }\left( \mathbf{H}\right) \left( k\right) =N_{\mathfrak{F}\mathbf{G} }\left( \mathfrak{F}\mathbf{H}\right) \left( k\right) ,\) provided \(N_{\mathbf{G}}\left( \mathbf{H}\right) \) is representable by a closed smooth subscheme of \(\mathbf{G.}\) Greenberg functor; linear algebraic groups; smooth group schemes; maximal tori Stasinski, A., Reductive group schemes, the Greenberg functor, and associated algebraic groups, J. Pure Appl. Algebra, 216, 1092-1101, (2012) Group schemes, Linear algebraic groups over arbitrary fields Reductive group schemes, the Greenberg functor, and associated algebraic groups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a complex abelian variety of dimension \(g\). An ample line bundle \(L\) on \(X\) defines a polarization \(H\) on \(X\), i.e., by taking its first Chern class, a Hermitean form \(H\) on the universal covering space of \(X\), whose imaginary part is integer-valued on the lattice defining \(X\). Assume that \(H\) is of type \(D=\text{diag}(d_ 1,\ldots,d_ g)\in M_ \mathbb{Z}(g,g)\) with respect to a Frobenius basis of the lattice. According to these data, one has the theta group \({\mathcal G}(L)\) of the line bundle \(L\) and the Heisenberg group \(Heis(D)\) of type \(D\). A theta structure for the line bundle \(L\) is then an isomorphism \(b:{\mathcal G}(L)\to Heis(D)\) inducing the identity on the natural subgroups \(\mathbb{C}^*\). If \(L\) is assumed to be a symmetric line bundle, i.e. \((- 1)^*_ XL\cong L\), then a symmetric theta structure on \(L\) is defined to be an ordinary theta structure which is compatible with the induced involutions on \({\mathcal G}(L)\) and \(Heis(D)\).
The aim of the present paper is to answer the natural question of whether a given line bundle admits a (symmetric) theta structure, and how many such structures exist. The authors give a complete and precise solution of this particular problem, which is of special importance for moduli problems in the theory of abelian varieties [cf. \textit{D. Mumford}, Invent. Math. 1, 287-354 (1966; Zbl 0219.14024)]. --- In section 1 of their paper, the authors introduce the concept of characteristics for an ample line bundle. These characteristics appear as elements in the universal covering space \(V\) of \(X\), and are generalizations of the classical theta characteristics for theta functions. Section 2 contains a slightly modified definition of the theta group of a line bundle, together with an explicit description of its elements via lifting to linear automorphisms of the (trivial) pull-back line bundle on the universal covering space \(V\). --- Heisenberg groups and theta structures for line bundles are briefly recalled in section 3. It is shown that every characteristic \(c\in V\) for a line bundle \(L\) determines a theta structure \(b_ c:{\mathcal G}(L)\to Heis(D)\) of \(L\), so that every ample line bundle on \(X\) really admits at least one theta structure. --- The precise number of theta structures on a given ample line bundle \(L\) is computed in section 4. More precisely, it is proved that this number is exactly \(h^ 0(X,L)^ 2\cdot\text{card}(Sp(D))\), where \(Sp(D)\) denotes the symplectic group of the polarization type \(D\). The proof is essentially based on the observation that the set of theta structures on \(L\) can be parametrized by the characteristics of \(L\), which had been introduced in section 1. --- The remaining two sections deal with symmetric line bundles and symmetric theta structures on them. The main result in setion 5 is the computation of the dimensions \(h^ 0(L)_ +\) and \(h^ 0(L)_ -\) of the eigenspaces of \(H^ 0(X,L)\) with respect to the involution \((-1)^*_ L\) induced by the normalized isomorphism \((- 1)_ L\) of the symmetric line bundle \(L\). The formula says that \(h^ 0(L)_ +={1\over 2}h^ 0(X,L)\) or \(h^ 0(L)_ -={1\over 2}h^ 0(X,L)\pm 2^{g-s-1}\), where \(s\) denotes the number of odd entries in \(D=\text{diag}(d_ 1,\ldots,d_ g)\), and in terms of the corresponding characteristic of \(L\) it is regulated which case occurs. In the concluding section 6, the authors compute the number of symmetric theta structures of a given symmetric ample line bundle \(L\). Depending on the characteristic of \(L\), the nunber \(n(L)\) of symmetric theta structures of \(L\) turns out to be equal to 0 or to \(\text{card}(Sp(D))\cdot 2^{2(g- s)}\). This means, in particular, there are exactly \(2^{2s}\) symmetric line bundles with polarization class \(H\), which admit symmetric theta structures. Moreover, putting the main results together, the following concluding theorem is obtained:
An ample symmetric line bundle \(L\) on \(X\) admits a symmetric theta structure if and only if the eigenspace dimensions \(h^ 0(L)_ +\) or \(h^ 0(L)_ -\) are maximal, and exactly \(2^{2s}\) out of \(2^{2g}\) line bundles in a given polarization class have this property.
The paper is very clearly and carefully written. The beautiful and important result is presented in a nearly self-contained way. abelian variety; polarization; theta group; Heisenberg group; moduli problems in the theory of abelian varieties; number of theta structures; symmetric line bundles; symmetric theta structures Birkenhake, Ch., Lange, H.: Symmetric theta-structures. Manuscr. Math.70, 67-91 (1990) Theta functions and abelian varieties, Algebraic theory of abelian varieties, Analytic theory of abelian varieties; abelian integrals and differentials Symmetric theta-structures | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 concerns constructing free resolutions of maximal Cohen-Macaulay modules (MCM) over complete intersection rings. These resolutions are often minimal. The results are used to characterize all MCM modules over complete intersections in terms of higher matrix factorizations. \par We provide a bit of historical background and context. The study of MCM modules over Cohen-Macaulay (CM) local rings is a generalization of the representation theory of finite dimensional algebras. In the first interesting case of a hypersurface, Eisenbud in [\textit{D. Eisenbud}, Trans. Am. Math. Soc. 260, 35--64 (1980; Zbl 0444.13006)] described MCM modules using matrix factorizations. This description arises from a study of the minimal free resolutions of the modules: for example, he showed that an MCM module with no free summands has a periodic resolution of period \(1\) or \(2\) and that these correspond to matrix factorizations of the defining equation. Matrix factorizations have since found applications in algebraic geometry, commutative and homological algebra, representation theory and physics amongst other fields. In [\textit{D. Orlov}, Mat. Contemp. 41, 75--112 (2012; Zbl 1297.14019)], a generalization of the theory of matrix factorizations to complete intersections was initiated, following which, other authors developed it further. However, these do not reveal the structure of minimal free resolutions of MCM modules. On the other hand, although there are many methods to construct free resolutions for wide classes of rings, these rarely yield minimal ones -- the knowledge of infinite minimal free resolutions of modules over rings is sparse. \par In this article (and in [\textit{D. Eisenbud} and \textit{I. Peeva}, Minimal free resolutions over complete intersections. Cham: Springer; (2016; Zbl 1342.13001)), the authors consider the general case of complete intersections. The case of codimension 2 was addressed in \textit{L. L. Avramov} and \textit{R.-O. Buchweitz}, J. Algebra 230, No. 1, 24--67 (2000; Zbl 1011.13007)] and [\textit{D. Eisenbud} and \textit{I. Peeva}, Acta Math. Vietnam. 44, No. 1, 141--157 (2019; Zbl 1419.13023)]. \par We now summarize the results in this paper. Let \(S\) be a Gorenstein local ring and \(M\) a finitely generated CM \(S\)-module of codimension \(c\). Fix a regular sequence \(\underline{f}:=f_1,\dots,f_c\in Ann(M)\) and set \(R:=S/(\underline{f})\). The authors inductively construct \textit{layered} \(S\)-free and \(R\)-free resolutions of \(M\) with respect to \(\underline{f}\). To do this, they use the theory of MCM approximations in the sense of [\textit{M. Auslander} and \textit{R.-O. Buchweitz}, Mém. Soc. Math. Fr., Nouv. Sér. 38, 5--37 (1989; Zbl 0697.13005)]. A brief review of these ideas are provided in section 2. More specifically, they describe codimension one MCM approximations in section 3. Using this, they obtain a \(2\)-term complex of free \(S\)-modules, \(\mathbf{B}^S\), and a map of complexes, \(\psi_{\bullet}^S:\mathbf{B}^S[-1]\rightarrow \mathbf{L'}\), where \(\mathbf{L'}\) is the inductively obtained \(S\)-free layered resolution of the essential MCM approximation of \(M\) over \(R':=R/(f_1,\dots,f_{c-1})\). In section \(4\), the layered \(S\)-free resolution of \(M\) is then constructed as the mapping cone of the map induced by \(\psi_{\bullet}^S\) between \(\mathbf{K}\otimes_S\mathbf{B}^S[-1]\rightarrow \mathbf{L'}\), where \(\mathbf{K}\) is the Koszul complex resolving \(R'\) over \(S\). For the \(R\)-free resolutions, the authors additionally make use of complete intersection (CI) operators and the Shamash construction -- these are reviewed in section \(5\). Let \(\mathbf{L'}\) now denote the inductively obtained \(R'\)-free layered resolution of the essential MCM approximation of \(M\) over \(R'\). Using the codimension one MCM approximations in section 3, a \(2\)-term complex of \(R'\)-free modules, \(\mathbf{B}\), and a map of complexes \(\psi_{\bullet}:\mathbf{B}[-1]\rightarrow \mathbf{L'}\) is obtained. In section \(6\), the \(R\)-free layered resolution of \(M\) is then constructed as the Shamash complex of the mapping cone of \(\psi_{\bullet}\). Note here that the mapping cone of \(\psi_{\bullet}\) is a \(R'\)-free resolution of \(M\). For high \(R\)-syzygies of a given \(R\)-module \(N\) of finite projective dimension over \(S\), these resolutions coincide with the ones constructed in [\textit{D. Eisenbud} and \textit{I. Peeva}, Minimal free resolutions over complete intersections. Cham: Springer (2016; Zbl 1342.13001)]. An alternate approach to the construction of the layered \(R\)-free resolutions is presented in section 9. It is obtained from a periodic exact sequence of \(R\)-modules that generalizes the \(R\)-free periodic resolution of a module over a hypersurface constructed in [\textit{D. Eisenbud}, Trans. Am. Math. Soc. 260, 35--64 (1980; Zbl 0444.13006)]. The ``layered'' terminology comes from the fact that these resolutions come with a natural filtration by subcomplexes, whose subquotients are the layers. \par Sections \(7\) and \(8\) concern the minimality of the layered resolutions constructed. In section \(7\), a criteria for minimality is provided in terms of the injectivity of certain CI operators (Theorem 7.1). This is then used to show under mild hypothesis that if \(N\) is a finitely generated MCM \(R\)-module of finite projective dimension over \(S\), then the layered resolutions over \(R\) and \(S\) with respect to \(\underline{f}\) of the \(n\)-th syzygy of \(M\) over \(R\) are minimal when \(n\geq 3+\text{max}\{c-2,r(\underline{f},N)\}\). Here \(r(\underline{f},N)\) is a function of the Castelnuovo-Mumford regularities over the rings of CI operators corresponding to the \(f_i\) of Ext modules involving the essential MCM approximations of \(N\) with respect to the \(S/(f_1,\dots,f_i)\). In particular, when \(S\) has an infinite residue field, the layered resolutions of sufficiently high \(R\)-syzygies of a given \(R\)-module \(N\) are minimal. Questions relating to the invariant \(r(\underline{f},N)\) are also explored in section \(8\). \par In section 10, an inductive definition of a CI matrix factorization is given (Definition 10.2) and is then used to define a CI matrix factorization module. The definitions here are essentially equivalent to the ones introduced by the authors in [\textit{D. Eisenbud} and \textit{I. Peeva}, Minimal free resolutions over complete intersections. Cham: Springer (2016; Zbl 1342.13001)]. These are then used to characterize all MCM modules over a complete intersection, generalizing the analogous result for hypersurfaces in [\textit{D. Eisenbud}, Trans. Am. Math. Soc. 260, 35--64 (1980; Zbl 0444.13006)]: A finitely generated \(R\)-module \(N\) is MCM if and only if it is a CI matrix factorization module for the sequence \(\underline{f}\). free resolutions; complete intersections; CI operators; Eisenbud operators; maximal Cohen-Macaulay modules Syzygies, resolutions, complexes and commutative rings, Cohen-Macaulay modules, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Complete intersections Layered resolutions of Cohen-Macaulay 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 We prove a general ``diagram method'' theorem valid for a quite large class of 3-folds with \(\mathbb{Q}\)-factorial singularities (see the basic theorems 1.3.2 and 3.2 and also theorem 2.2.6). This generalizes our results on Fano 3-folds with \(\mathbb{Q}\)-factorial terminal singularities [\textit{V. V. Nikulin}, J. Math. Kyoto Univ. 34, No. 3, 495-529 (1994; Zbl 0839.14030)]. -- As an application, we get the following result about Calabi-Yau 3-folds \(X\):
Assume that the Picard number \(\rho(X)>40\). Then one of two cases (i) or (ii) holds:
(i) There exists a small extremal ray on \(X\).
(ii) There exists a nef element \(h\) such that \(h^3=0\) (thus, the nef cone \(\text{NEF}(X)\) and the cubic intersection hypersurface \({\mathcal W}_X\) have a common point; here, we do not claim that \(h\) is rational!).
As a corollary, we get: Let \(X\) be a Calabi-Yau 3-fold. Assume that the nef cone \(\text{NEF}(X)\) is finite polyhedral and \(X\) does not have a small extremal ray. Then there exists a rational nef element \(h\) with \(h^3=0\) if \(\rho (X)>40\).
To prove these results on Calabi-Yau manifolds, we also use a result of the appendix by V. V. Shokurov on the length of divisorial extremal rays. Thus the above results on Calabi-Yau 3-folds are joint work with V. V. Shokurov. We also discuss the generalization of the above results to so-called \(\mathbb{Q}\)-factorial models of Calabi-Yau 3-folds. This sometimes enables us to play the same game even when the Mori cone is not polyhedral, or when there are small extremal rays. effective 1-cycles; length of extremal ray curve; Kähler cone; Calabi-Yau 3-folds; Picard number; nef cone; length of divisorial extremal rays Viacheslav V. Nikulin, The diagram method for 3-folds and its application to the Kähler cone and Picard number of Calabi-Yau 3-folds. I, Higher-dimensional complex varieties (Trento, 1994) de Gruyter, Berlin, 1996, pp. 261 -- 328. With an appendix by Vyacheslav V. Shokurov. Calabi-Yau manifolds (algebro-geometric aspects), Minimal model program (Mori theory, extremal rays), \(3\)-folds The diagram method for 3-folds and its application to the Kähler cone and Picard number of Calabi-Yau 3-folds. I. -- With an appendix by Vyacheslav V. Shokurov: Anticanonical boundedness for 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 \((X^n, \check{X}^n)\) be a mirror pair of an \(n\)-dimensional complex torus \(X^n\) and its mirror partner \(\check{X}^n\). Then, a simple projectively flat bundle \(E(L,\mathcal{L})\rightarrow X^n\) is constructed from each affine Lagrangian submanifold \(L\) in \(\check{X}^n\) with a unitary local system \(\mathcal{L}\rightarrow L\). In this paper, we first interpret these simple projectively flat bundles \(E(L,\mathcal{L})\) in the language of factors of automorphy. Furthermore, we give a geometric interpretation for exact triangles consisting of three simple projectively flat bundles \(E(L,\mathcal{L})\) and their shifts by focusing on the dimension of intersections of the corresponding affine Lagrangian submanifolds \(L\). Finally, as an application of this geometric interpretation, we discuss whether such an exact triangle on \(X^n (n\geq 2)\) is obtained as the pullback of an exact triangle on \(X^1\) by a suitable holomorphic projection \(X^n\rightarrow X^1\). Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Mirror symmetry (algebro-geometric aspects) Geometric interpretation for exact triangles consisting of projectively flat bundles on higher dimensional complex tori | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \((R, \mathfrak{m}, k)\) denote a local Cohen-Macaulay ring such that the category of maximal Cohen-Macaulay \(R\)-modules \(\mathbf{mcm} R\) contains an \(n\)-cluster tilting object \(L\). In this paper, we compute the Quillen \(K\)-group \(G_1(R) := K_1(\mathbf{mod} R)\) explicitly as a direct sum of a finitely generated free abelian group and an explicit quotient of \(\text{Aut}_R(L)_\text{ab}\) when \(R\) is a \(k\)-algebra and \(k\) is algebraically closed with characteristic not two. Moreover, we compute \(\text{Aut}_R(L)_\text{ab}\) and \(G_1(R)\) for certain hypersurface singularities. Cohen-Macaulay; \(n\)-cluster tilting; \(K\)-theory; hypersurface singularities; automorphism groups Cluster algebras, Representations of orders, lattices, algebras over commutative rings, Module categories in associative algebras, Cohen-Macaulay modules, Singularities in algebraic geometry, Representations of associative Artinian rings \(G\)-groups of Cohen-Macaulay rings with \(n\)-cluster tilting objects | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(R\) be a commutative ring and \(A\) a generalized Cartan matrix. Denote by \(\mathfrak{St}_A(R)\) the Steinberg group of type \(A\) as defined by \textit{J. Morita} and \textit{U. Rehmann} [Tohoku Math. J. (2) 42, No. 4, 537--560 (1990; Zbl 0701.19001)]. Recall that this group is generated by elementary root unipotents \(x_\alpha(t)\), parametrized by all real roots of the root system \(\Phi=\Phi(A)\), subject to Chevalley relations which are imposed for all prenilpotent pairs of real roots.
In the paper under review a new group \(\mathfrak{PSt}_A(R)\), called \textit{pre-Steinberg group}, is introduced. This group has the same generators as \(\mathfrak{St}_A(R)\), but the Chevalley relations are imposed only for classically prenilpotent pairs of roots.
It turns out that this group coincides with \(\mathfrak{St}_A(R)\) under mild assumptions on the form of the Dynkin diagram of \(A\) (e.\,g. this is true if every subdiagram with \(\leq 3\) vertices is a Dynkin diagram of a finite root system, see Theorem~1.1). The key virtue of the group \(\mathfrak{PSt}_A(R)\) is that it admits a very compact presentation formulated solely in terms of the Dynkin diagram of \(A\), in which, moreover, no signs are left implicit, see Theorem~1.2. Another advantage of this presentation is that its very form implies the fact that the group \(\mathfrak{PSt}_A(R)\) is a colimit of groups \(\mathfrak{PSt}_B(R)\), where \(B\) runs over all \(1\times 1\) and \(2\times 2\) Cartan submatrices of \(A\) (this fact is a generalization of the classical Curtis-Tits presentation known in the theory of finite groups of Lie type).
Based on the Curtis-Tits property of pre-Steinberg groups and \textit{S. Splitthoff}'s earlier result on finite presentation of Steinberg groups (see~[Contemp. Math. 55, 635--687 (1986; Zbl 0596.20034)]) the author deduces finite presentability theorems for a large class of pre-Steinberg groups and Tits' Kac-Moody groups over rings satisfying some finite presentability assumptions, see Theorem~1.4 and Corollary~1.5, respectively. Kac-Moody group; Curtis-Tits presentation; Steinberg group; pre-Steinberg group D. Allcock, \textit{Steinberg groups as amalgams}, in preparation. Steinberg groups and \(K_2\), Kac-Moody groups, Group schemes, Generators, relations, and presentations of groups, Linear algebraic groups over adèles and other rings and schemes Steinberg groups as amalgams | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 complex analytically irreducible quasi-ordinary (q.o) singularity, defined by \(f\in \mathbb C\{x_1,x_2\}[z]\). It can be parametrized in the form \(x_1=x_1\), \(x_2=x_2\), \(z=\zeta(x_1,x_2)\) with \(\zeta\in \mathbb C\{x_1,x_2\}[z]\). [\textit{Y.-N. Gau}, Mem. Am. Math. Soc. 388, 109--129 (1988; Zbl 0658.14004)] as shown: A finite set of exponents in the support of the series \(\zeta\) -- they are called the characteristic exponents -- are complete invariants of the topological type of the singularity.
In the paper under review, the authors look for invariants for {\em all} types of singularities. They consider the set of jet schemes of \(X\). For \(m\in\mathbb N\), they define a functor \(F_m\colon \mathbb C\text{-Schemes}\to \text {Sets}\) which is representable by a \({\mathbb C}\)-scheme \(X_m\), the \(m\)th jet scheme. There is a canonical projection \(\pi_m \colon X_m\to X\). In section 4 q.o.\ surfaces with one characteristic exponent are considered. The irreducible components of the \(m\)-jet schemes through the singular locus of a such a surface are described in Th.\ 4.14. A graph \(\Gamma\) is constructed which represents the decomposition of \((\pi^{-1}_m(X_{\text{Sing}}))_{\text{red}}\) for every \(m\). The graph \(\Gamma\) is equivalent to the topological type of the singularity. In section 5 these results are generalized to the general case. quasi-ordinary; surface singularities; jet schemes Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties Jet schemes of quasi-ordinary surface singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities A sandwiched singularity \(X\) of a surface is a normal surface singularity which dominates birationally a non singular surface \(S\). These singularities are obtained from the basis \(S\) by blowing-up a complete \(\mathfrak M\)-primary ideal \(I\) of the local ring \({\mathcal O}_{S,O}\), \(S\) is a complex regular analytic surface. The usual problem in this theory is to get all possible information from \(I\subset {\mathcal O}_{S,O}\) on the singularities of \(X\). Here the informations given by the author are the number of singularities on \(X\), their fundamental cycles and multiplicities.
The main theorem 3.5 gives, in the more general case where \({\mathcal O}_{S,O}\) is a local \(\mathbb C\)-algebra having a rational singularity, a bijection between the set of complete ideals of codimension 1 (as \(\mathbb C\)-vector space) contained in \(I\) and the set of points in the exceptional locus of the surface \(X=\text{Bl}_I(R)\).
In the section 4, the author applies his theorem to the case where \(S\) is regular and \(X\) a sandwiched singularity. To every exceptional point \(Q\in X\), there is the associated ideal \(I_Q\) of 3.5 and to \(J\) a complete \(\mathfrak M\)-primary ideal, the author associates the weighted cluster of base points of \(J\) (base points are closed points in \(X'\) where general elements of \(J\) go through, \(X'\to X\) is any sequence of blow-ups of closed points).
All this leads to an explicit formula in theorem 4.7, which gives the multiplicity of an exceptional point \(Q\in X\) in terms of the clusters given by the base points of \(I\) and of \(I_Q\). regular; complete ideals; sandwiched singularity; analytic surface Fernández-Sánchez, J.: On sandwiched singularities and complete ideals. J. pure appl. Algebra 185, 165-175 (2003) Singularities of surfaces or higher-dimensional varieties, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Rational and birational maps, Complex surface and hypersurface singularities On sandwiched singularities and complete ideals. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \((Y,P)\) be a threefold singularity of index \(1\), namely such that the canonical divisor \(K_Y\) is Cartier, and let \(\pi:X\to Y\) be a resolution: One can write \(K_X=\pi^*K_Y+ \sum a_i E_i\), where the \(E_i\)'s are the exceptional divisors of \(\pi\). The singularity \((Y,P)\) is \textit{terminal} if \(a_i>0\) for every \(i\), and the \textit{minimal discrepancy} of \(\pi\) is the minimum of the coefficients \(a_i\). It is known [\textit{M. Reid} in: Algebraic Varieties and Analytic Varieties, Proc. Symp., Tokyo 1981, Adv. Stud. Pure Math. 1, 131-180 (1983; Zbl 0558.14028)] that the terminal isolated singularities of index 1 are precisely the cDV (compound DuVal) singularities, namely singularities analytically equivalent to hypersurface singularities such that the generic linear section through the singular point is a rational double point.
Using this explicit geometric description, the author proves that every isolated terminal threefold singularity of index \(1\) admits a resolution having minimal discrepancy equal to \(1\). This result has been generalized to the case of terminal singularities of index \(r\), namely such that \(r\) is the smallest integer such that \(rK_Y\) is Cartier [see \textit{Y. Kawamata}'s appendix (p. 201-203) to \textit{V. V. Shokurov}'s paper in Russ. Acad. Sci., Izv., Math. 40, No. 1, 95-202 (1993); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 56, No. 1, 105-203 (1992; Zbl 0785.14023)]. threefold singularity; terminal isolated singularities; compound DuVal singularities; minimal discrepancy Markushevich, D., \textit{minimal discrepancy for a terminal cdv singularity is 1}, J. Math. Sci. Univ. Tokyo, 3, 445-456, (1996) Singularities in algebraic geometry, \(3\)-folds, Minimal model program (Mori theory, extremal rays) Minimal discrepancy for a terminal cDV singularity is 1 | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(\Gamma\) be a finite subgroup of \(\text{SL}_2(\mathbb{C})\). In this paper, the author considers \(\Gamma\)-fixed point sets in Hilbert schemes of points on the affine plane \(\mathcal{C}^2\). The direct sum of homology groups of components has a structure of a representation of an affine Lie algebra \(\widehat{\mathfrak g}\) corresponding to \(\Gamma\). If one replaces homology groups by equivariant \(K\)-homology groups, one gets a representation of the quantum toroidal algebra \({\mathbf{U}}_q({\mathbf L}{\widehat{\mathfrak g}})\). He also discusses a higher rank generalization and character formulas in terms of intersection homology groups. affine Lie algebra; quantum toroidal algebra; Hilbert schemes; quiver varieties Nakajima, H.: Geometric construction of representations of affine algebras. In: Proceedings of the International Congress of Mathematicians, Volume I, 2003, pp 423--438 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 Geometric construction of representations of 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 \textit{C. De Concini} and \textit{C. Procesi} [Lect. Notes Math. 996, 1--44 (1983; Zbl 0581.14041)] invented the \textit{Wonderful Compactification} of a complex semisimple adjoint group \(G\). This compactificaton is a smooth projective variety containing \(G\) as a dense open subvariety and where the boundary is a normal crossing divisor with structure determined by the root datum of \(G\). This compactification is claimed to have many applications, particularly in spherical geometry.
In the present article, the author consider the loop group \(LG\) of the group \(G\) and construct an analogue of the wonderful compactification. The loop group can be defined as the group of maps from a punctured formal disc to the group \(G\). The points are the \(\mathbb C((z))\)-points in \(G\), that is \(G(\mathbb C((z))\), where \(C((z))\) is the field of formal Laurent series. An embedding \(X^{\mathrm{aff}}\) is constructed which is not compact, but which have nice properties justifying it as a loop group analogue of the wonderful compactification.
The main application of the theory developed in this work concerns the moduli stack \(\mathcal M_G(C)\) of \(G\)-bundles on a family of nodal curves \(C\). This is not a compact stack, and so \(X^{\mathrm{aff}}\) is used as a kind of compactification. The compactification of the coarse moduli space of \(G\)-bundles on \(C\), is the original goal of the wonderful compactification. There exists compactification for semistable groups, but the author claims that non of these are leading to a satisfactory construction of a compact moduli space for bundles over families of nodal curves.
This article shows that a compactification \(\overline G\) of \(G\) is insufficient to compactify \(\mathcal M_G(C)\) in general, but that the embedding \(X^{\mathrm{aff}}\) gives enough additional data to compactify \(\mathcal M_G(C)\).
\textit{E. Frenkel} et al. [Adv. Math. 288, 201--239 (2016; Zbl 1342.14111)] use the compactification of the moduli of \(\mathbb C^\times\)-bundles to define Gromov-Witten invariants for \([\mathrm{pt}/\mathbb C^\times]\). Their index theorem suggest that similar invariants can be defined on a completion of \(\mathcal M_G(C)\). Then the parallels between \(X^{\mathrm{aff}}\) and the wonderful compactification \(\overline{G_{\mathrm{ad}}}\) suggest that also other constructions related to \(\overline{G_{\mathrm{ad}}}\) have loop analogues.
Vinberg has given an alternative construction of \(\overline{G_{\mathrm{ad}}}\) using a monoid \(S_G\) called the Vinberg monoid, the construction was generalized by Thaddeus and Martens to provide stacky compactifications for any split reductive group. The main result of this article makes it relevant to ask if there is a loop group analogue of the Vinberg monoid.
In geometric representation theory, the wonderful compactification is used e.g. to define the Harish-Chandra transform defined on the category of \(D\)-modules on \(G\), and the results of this article indicates that similar constructions exists for \(D\)-modules on the loop group LG. The wonderful compactification generalizes to elliptic Springer theory by giving an analogue of Lustig's character sheaves for loop groups. This is originally used to give a geometric construction of character sheaves for \(p\)-adic groups. In the present work, the intention is to study sheaves on conjugacy classes in \(LG\). Applying results from \textit{V. Baranovsky} and \textit{V. Ginzburg} [Int. Math. Res. Not. 1996, No. 15, 733--751 (1996; Zbl 0992.20034)], this can be studied by sheaves on the moduli space \(M_{G,E}\) of semistable bundles on an elliptic curve.
Because conjugacy classes in \(LG\) are \(\Delta(LG)\) orbits in \(LG\), the author investigates if \(\Delta(\mathbb C^\times\ltimes LG)\) orbits in \(X^{\mathrm{aff}}\) have a modular interpretation like the interpretation given by Baranovsky and Ginsburg. If this was true, then it can be used to formulate a theory of character sheaves for loop groups.
The author points out that \(LG\) is an ind-scheme rather than a scheme. In this article, the objects under study is thus the category of \textit{ind-schemes}, and the reader is supposed to have good knowledge of such, together with their stack theory. Then \(LG\) is studied through its representations, and it is proved that \(LG\) has a class of projective representations behaving in many ways like the finite dimensional representations of a semisimple group. These are the \textit{honest representations} of a central extension \(\widetilde{LG}\) of \(LG\) by \(\mathbb C^\times\). Such representations are infinite dimensional, and one introduce another \(\mathbb C^\times\) and puts \(G^{\mathrm{aff}}=\mathbb C^\times\ltimes\widetilde{LG}\) to decompose the representation into a direct sum of finite dimensional weight spaces for a maximal torus in \(G^{\mathrm{aff}}(\mathbb C)\). The author uses the representation theory of \(G^{\mathrm{aff}}(\mathbb C)=\mathbb C^\times\ltimes\widetilde{LG}(\mathbb C)\) to replace representation theory of a finite dimensional semisimple group. \(G^{\mathrm{aff}}(\mathbb C)\) is called a Kac-Moody group and is associated to an affine Dynkin diagram in the same way a semisimple group is associated to a Dynkin Diagram.
The wonderful compactification of \(G\) is the compactification of \(G_{\mathrm{ad}}=G/Z(G)\) given by choosing a regular dominant weight \(\lambda\), letting \(V(\lambda)\) be the associated highest weight representation of \(G\), and defining \(\overline{G_{\mathrm{ad}}}=\overline{G\times G[\mathrm{id}]}\subset\mathbb P \mathrm{End}(V(\lambda)).\) The analogue construction is then given by letting \(\lambda\) be a dominant weight \(\underline{\lambda}\) of \(G^{\mathrm{aff}}\) and using the associated representation \(V(\underline{\lambda})\) to construct an ind-scheme \(\mathbb P \mathrm{End}^{\mathrm{ind}}(V(\underline{\lambda}))\) to obtain \(X^{\mathrm{aff}}=\overline{G^{\mathrm{aff}}\times G^{\mathrm{aff}}[\mathrm{id}]}\subset\mathbb P \mathrm{End}^{\mathrm{ind}}(V(\underline{\lambda})).\)
The main result in the article states that when \(G\) is a simple, connected and and simply connected group over \(\mathbb C\) with maximal torus \(T\), the ind-scheme \(X^{\mathrm{aff}}\) contains \(G^{\mathrm{aff}}_{\mathrm{ad}}\) as a dense open sub-ind scheme. The main theorem also contains statements of the properties of \(X^{\mathrm{aff}}\), the boundary of the wonderful compactification of \(LG\), and the properties of the maximal torus. Also, the maximal tori are studied by toric varieties, leading to an explicit description of the orbits of the group-action.
The article contains a very nice study of compactifications, and introduce new tools to study such. It is not very self-contained, as stacks, ind-schemes, and Lie-algebra actions is a basis for the article, making it much deeper than it seems. However, taking this basis into account, the article is very well written and give a framework for wonderful compactifications. loop groups; affine Lie algebras; moduli of \(G\) bundles on curves; embeddings of reductive groups; representation theory; spherical varieties; wonderful compactification; torus group; Harish-Chandra transform; character sheaves; ind-scheme; compactification; flag varieties; divisors in ind-schemes Solis, P., A wonderful embedding of the loop group Compactifications; symmetric and spherical varieties, Representations of Lie and linear algebraic groups over real fields: analytic methods A wonderful embedding of the loop 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 It is known that every topological nature of a normal complex surface singularity is described in terms of the weighted dual graph of the exceptional set a good resolution.
The main result of this paper is that for a given topological type \(\Gamma\) of a normal surface singularity, though the families of generic hyperplane sections and that of polar curves of the generic plane projections of a singularity with graph \(\Gamma\) depend on the complex analytic type, the topological types of them are finite. In other words, the number of the weighted dual graph for the minimal good resolution which factors through the blowup of the maximal ideal and the Nash transform of a singularity is finite.
One of key points for the proof is that the upper bounds of the multiplicity and the polar multiplicity, which are numerical invariants associated with generic hyperplane sections and the generic plane projections, are given by \(\Gamma\). Many arguments in the proof rely on [\textit{A. Belotto da Silva} et al., Geom. Topol. 26, No. 1, 163--219 (2022; Zbl 1487.32166)] and related results; however, this article includes concise explanations for those ingredients. complex surface singularities; polar curves; hyperplane sections; Nash transform; Lipschitz geometry; multiplicity; Mather discrepancy Singularities in algebraic geometry, Complex surface and hypersurface singularities, Modifications; resolution of singularities (complex-analytic aspects), Global theory and resolution of singularities (algebro-geometric aspects) Polar exploration of complex surface germs | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(C\) be an irreducible nodal curve of arithmetic genus 2. The author gives a geometrical description of the moduli space \(\text{SU}(2, \omega_C)\) of semistable rank 2 vector bundles on \(C\) with determinant \(\omega_C\). When \(C\) is smooth, this moduli space was studied by \textit{M. S. Narasimhan} and \textit{S. Ramanan} [Ann. Math., II. Ser. 89, 14-51 (1969; Zbl 0186.54902)] who showed that it is naturally isomorphic to the 3-dimensional linear system \(|2\Theta|\), where \(\Theta\) is a symmetric theta divisor on the Jacobian \(J(C)\).
In the nodal case, the author shows that \(\text{SU}(2,\omega_C)\) is isomorphic to an open set of the linear system of quadrics containing a convenient projective model of \(C\). This open set consists of the quadrics whose singular locus does not contain any node of \(C\). The proof is based on a construction considered by \textit{S. Brivio} and \textit{A. Verra} [Duke Math. J. 82, No. 3, 503-552 (1996; Zbl 0876.14024)]. More precisely, let \(t\in\text{Pic}^2C\) and \(L:=\omega_C(2t)\). \(|L|\) embeds \(C\) into \(\mathbb{P}(H^0(L)) \simeq\mathbb{P}^4\). Let \(E\) be a semistable 2-bundle on \(C\) with \(\wedge^2E\simeq L\). Then \(h^0(E)=4\) and the evaluation map \(H^0(E)\otimes{\mathcal O}\to E\) defines a linear rational map \(\mathbb{P}(H^0(L)) \to\mathbb{P}(\wedge^2 H^0(E))\). The inverse image of the Plücker quadric in \(\mathbb{P}(\wedge^2H^0(E))\) is a quadric in \(\mathbb{P}(H^0(L))\) containing the embedding of \(C\).
When \(C\) is smooth one recuperates the result of Narasimhan and Ra-anan because, as it is shown in section 4 of the above mentioned paper of Brivio and Verra, the linear system of quadrics containing the embedding of \(C\) into \(\mathbb{P}(H^0(L))\) is naturally isomorphic to \(|2\Theta|\). nodal curve; theta divisor; Jacobian Brivio S. (1998). On rank 2 semistable vector bundles over an irreducible nodal curve of genus 2. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 1(3): 611--629 Singularities of curves, local rings, Vector bundles on curves and their moduli, Jacobians, Prym varieties, Theta functions and abelian varieties, Theta functions and curves; Schottky problem On rank 2 semistable vector bundles over an irreducible nodal curve of genus 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 Main purpose of the present article is to show the existence of a coarse moduli space of irreducible connections on vector bundles of rank \(2\) on \(\mathbb{P}^1(\mathbb{C})\) having regular singularities in three fixed points, say, \(\{0,1, \infty \}\). Put \(U_0 = \mathbb{P}^1(\mathbb{C})- \{\infty \}= \mathbb{C}\) with coordinate \(z\) and \(U_{\infty}=\mathbb{P}^1(\mathbb{C})-\{0\}=\mathbb{C}\) with coordinate \(x\) such that \(xz=1\). Let \(S=\{s_1,s_2,\ldots,s_n\}\) be a finite collection of points of \(X=P^1(\mathbb{C})\), \(n\geq 2\), \(s_i\ne s_j\), if \(i\ne j\), \(z(s_i)=a_i\). Let \(\mathcal E\) be a holomorphic vector bundle of rank \(2\) on \(X= \mathbb{P}^1(\mathbb{C})\). A connection \(\nabla\) on \(\mathcal E\) having regular singularities on \(S\) is, by definition: (1) \(\nabla\) is a holomorphic connection of \({\mathcal E}_{| X-S}\) (i.e. a \({\mathbb{C}}\)- linear map \(\nabla: \mathcal E_{| X-S}\to \Omega^1(\mathcal E_{| X-S})\) satisfying \(\nabla(fe) = df\otimes e+f\nabla(e)\) for local sections \(f\) and \(e\) of \(\mathcal O_X\) and \(\mathcal E\), respectively); (2) for any \(s\in S\), there exist an open neighbourhood \(U\) of \(s\) in \(X\), a base \(\omega\) of \(\Omega^1_U\) and a base \((e)\) of \(\mathcal E_{| U}\) meromorphic in \(s\) (i.e. there exists an element \(T\in \mathrm{GL}(2,\Gamma(U,\mathcal O_ X(S)))\) and a base \((g)\) of \(\mathcal E_{| U}\) with \((e)=(g)T\) on \(T-\{s\}\) such that if we write \(\nabla(e) = M\cdot(\omega \otimes e)\), then each component of the matrix \(M\) has a pole of order at most one at \(s\) and holomorphic on \(U-\{s\}\).
Put \(_z\nabla(v) = <\nabla(v),d/dz>\), \(_x\nabla(v)=<\nabla(v),d/dx>\), where \(v\) is a local section of \(\mathcal E\). We have \(_z\nabla = -x^2_x\nabla\) on \(U_0\cap U_{\infty}- S\). The author first shows the following theorem, which enable him to reduce the moduli problem of connections to that of pairs of matrices:
Theorem 1.2.1. Let \(\mathcal E\) and \(S\) be the same as above and \(\nabla\) be a connection on \(\mathcal E\) having regular singularities in \(S\). Then there exists a basis of \(\mathcal E\) meromorphic in \(S\) such that with respect to this basis, if \(S\subset U_0\), then
\[ _z\nabla =(d/dz)+(A_1/(z- a_1))+ \ldots +(A_n/(z-a_n)),\]
where \(A_1,\ldots,A_n\) are \(2\times 2\) complex matrices and \(A_1+ \ldots +A_n=0\).
The author shows that the same theorem holds for algebraic vector bundles \(\mathcal E\) on \(P^1(K)\) and an algebraic connection with regular singularities on \(S\), if \(K\) is an algebraically closed field of characteristic \(0\). He also shows that if \(K\) is not algebraically closed with \(\mathrm{char}(K)=0\), then the theorem does not necessarily hold. (He gives the necessary and sufficient conditions that the theorem holds in this case: Theorem 1.3.6.)
Let \(P_m\) be the set consisting of pairs \((A_1,A_2)\) of \(K\)-valued \(2\times 2\) matrices. The author defines two equivalence relations \(\sim,\approx\) on \(P_m\) by:
\((B_1,B_2)\sim(A_1,A_2)\) if \((B_1,B_2)=(T^{-1}A_1T,T^{-1}A_ 2T)\) for some \(T\in \mathrm{GL}(2,K)\);
\((B_1,B_2)\approx(A_1,A_2)\) if \((B_1/z)+(B_2/(z-1)) = T^{-1}((A_1/z)+(A_2/(z-1)))+T^{-1}dT/dz\) for some \(T\in \mathrm{GL}(2,K[z,1/z,1/(z- 1)])\).
Classifying elements of \(P_m\) relative to the first equivalence relation \(\sim\) agrees with classifying Fuchsian systems and classifying elements of \(P_m\) relative to the second equivalence relation \(\approx\) agrees with classifying connections (Theorem 1.2.1 and Lemma 3.1.5).
In {\S} 2, the author classifies certain subsets of Fuchsian systems. Let \(V\) be the category of analytic spaces (or algebraic varieties). For any \(S\in V\), a family of pairs of matrices over \(S\) is a triple (\(\mathcal E,A_1,A_2)\) consisting of a vector bundle \(\mathcal E\) of rank \(2\) on \(S\) and endomorphisms \(A_1,A_2\) of \(\mathcal E\). Two families (\(\mathcal E,A_1,A_2)\) and (\(\mathcal E',A'_1,A'_2)\) on \(S\) are called equivalent, if there exist an open covering \(\{U_j\}_{j\in J}\) of \(S\) and isomorphisms \(\phi_j: \mathcal E_{| U_ j}\to \mathcal E'_{| U_ j}\) such that \(A'_{k| U_ j}=\phi_ j(A_{k| U_ j})\phi_ j^{-1}, k=1,2\), for all j. We denote the equivalence class of (\(\mathcal E,A_1,A_2)\) by \(c(\mathcal E,A_1,A_2)\). By \(F_m\) we mean a contravariant functor from \(V\) to \((Set)\) defined by \(F_m(S)=\{c(\mathcal E,A_1,A_2)\}\). Similarly one defines a subfunctor \(F\) of \(F_m\) by \(F(S)=\{c(\mathcal E,A_1,A_2)| (A_1(s),A_2(s))\) is irreducible, that is, \(A_1(s)\) and \(A_2(s)\) have no common eigenvectors in \(\mathcal E(s)\) for any \(s\in S\}\). The author shows that \(F_m\) has no coarse moduli space (proposition 2.2) but \(F\) has a coarse moduli space (Theorem 2.3.2). Here, a coarse moduli space for a functor \(G\) from \(V\) to \((Set)\) is a pair \((N,\Phi)\) with \(N\in V\), \(\Phi: G\to h_N = \Hom(N,\cdot)\) such that \(\Phi(p): G(p)\to h_N(p)\) is bijective where \(p\) is a point and for each \(L\in V\) and each morphism \(\Psi: N\to h_L,\) there is a unique morphism \(f: N\to L\) such that the diagram \(G\to^{\Phi}h_N\to^{h_f}h_L; G\to^{\Psi}h_L\) is commutative.
In {\S} 3 the author classifies the irreducible connections on holomorphic vector bundles of rank \(2\) on \(\mathbb{P}^1(\mathbb{C})\) having regular singularities in three fixed points \(0,1,\infty\). A family of irreducible connections on an analytic space \(S\) is, by definition, a pair (\(\mathcal E,\nabla)\) where \(\mathcal E\) is a vector bundle of rank 2 on \(S\times \mathbb{P}^1(\mathbb{C})\) and \(\nabla\) is an irreducible relative connection on \(\mathcal E\) having regular singularities in \(\{0,1,\infty \}\). (''Irreducible'' means that the family of pairs \((A_1,A_2)\) on \(S\) associated with \(\nabla\) by Theorem 1.2.1 is irreducible.) Two pairs (\(\mathcal E,\nabla)\) and (\(\mathcal E',\nabla')\) are called equivalent if for each \(x\in X=S\times \mathbb{P}^1(\mathbb{C})\) there exist a neighbourhood \(U\) of \(x\) in \(X\) and an isomorphism \(\phi: E_{| U}\to \mathcal E'_{| U}\) meromorphic along \(Y=S\times \{0,1,\infty \}\) such that on \(U-Y\) we have \(_z\nabla =_z\nabla'\). Then the author shows that the functor \(G\) from \(V\) to \((Set)\) defined by \(G(S)= \) the set of equivalence classes of (\(\mathcal E,\nabla)\) has a coarse moduli space. Fuchsian system; coarse moduli space of irreducible connections on vector bundles of rank 2; regular singularities Algebraic moduli problems, moduli of vector bundles, Connections (general theory), Complex-analytic moduli problems, Structure of families (Picard-Lefschetz, monodromy, etc.), Sheaves in algebraic geometry, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Other connections Moduli spaces for pairs of \(2\times 2\)-matrices and for certain connections on \(\mathbb{P}^1(\mathbb{C})\) | 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 paper under review is devoted to the study of Artin group actions on the bounded derived category of coherent sheaves \(D^b(X)\) on a complex quasiprojective threefold \(X\) arising by glueing minimal resolutions of quotient singularities \(\mathbb{C}^2/\Gamma\) along a smooth curve \(B\), \(\Gamma\) being a finite subgroup of \(\text{SL}(2,{\mathbb{C}})\). Issues similar to this were investigated e.g. \textit{P. Seidel} and \textit{R. Thomas}, [Duke Math. J. 108, 37--108 (2001; Zbl 1092.14025)] and by \textit{R.P. Horja} [Hypergeometric functions and mirror symmetry in toric varieties, \texttt{math.AG/9912109}; Derived category automorphisms from mirror symmetry; \texttt{math.AG/0103231}]. However, braid group actions constructed by Seidel and Thomas do not necessarily agree with those considered here, as noted by the author in [Comm. Math. Phys. 238, 35--51 (2003; Zbl 1037.81083)]. Given the group \(\Gamma\), one considers its McKay graph, which is an extended diagram of Dynkin type, and its associated simply laced diagram \(\Delta\) (i.e. a Dynkin diagram of type \(A\,D\,E\)), having Cartan algebra \({h}_{\Delta}\). This diagram describes the configuration of exceptional divisors in the minimal resolution \(Y\) of \({\mathbb{C}}^2/\Gamma\). Considering the normalizer \(N_{\Gamma}\), a cocycle \(\alpha \in {H}^1(B_{\text{ét}},N_{\Gamma})\), and its image in \({H}^1(B_{\text{ét}},\text{GL}({h}_{\Delta}))\), one obtains a bundle \(\mathcal{H}\) on \(B\). Also, \(\alpha\) gives a subgroup \(A\) of \(\text{Aut}(\Delta)\) and a quotient diagram \(\Xi = \Delta / A\), which is again of Dynkin type. The threefold \(X\) obtained glueing over \(\alpha\) the resolutions \(Y \times B_l \rightarrow {\mathbb{C}}^2/\Gamma \times B_l\) contains a configuration of geometrically ruled surfaces described by the diagram \(\Xi\). Moreover, pulling back the universal deformation of \(Y\) one constructs a smooth family \(\mathcal{X}\) of threefolds fibered over \(B\), with the vector space \(T={H}^0(B,\mathcal{H})\) as base, whose central fiber is \(X\). The Artin group \(B_{\Xi}\) associated to \(\Xi\) has a generator \(R_i\) for each vertex of \(\Xi\) and a relation of the form \(R_i\,R_j \ldots = R_j \, R_i \ldots\), with \(m_{i,j}\) elements in each side of the relation, \(m_{i,j} \in \{2,3,4,6\}\) being the label of the pair \((i,j)\). The group \(B_{\Xi}\) covers the Weil group \(W_{\Xi}\) and equals the standard braid group if \(\Xi\) is of type \(A_n\). The group \(W_{\Xi}\) acts on \(T\) by its action \({h}_{\Delta}\). The main result of this paper asserts that the group \(B_{\Xi}\) acts on \(D^b(\mathcal{X})\) by relative autoequivalences over \(W_{\Xi}\)-invariant finite--dimensional families. A generator \(B_i\) of \(B_{\Xi}\) corresponds to a kernel in \(D^b(\mathcal{X \times X})\), which is defined fiberwisely on \(X_s\) as the universal perverse coherent point sheaf, relative to a contraction of \(X_s\) associated to the vertex \(i\) of \(\Xi\). The action is defined also on the central fiber \(X\), and the author proves it to be faithful in case \(\Xi\) is of type \(A_n\) or \(C_n\). A projective example is also discussed, where a similar result takes place. Artin group actions; derived categories; McKay correspondence; Fourier-Mukai functors; Dynkin diagrams; universal deformation of threefolds; derived autoequivalences Szendroi, Balázs, Artin group actions on derived categories of threefolds, J. Reine Angew. Math., 572, 139-166, (2004) \(3\)-folds, Calabi-Yau manifolds (algebro-geometric aspects), Derived categories, triangulated categories, Braid groups; Artin groups Artin group actions on derived categories of 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 The Gross-Siebert program is an algebraic approach to the construction of mirror varieties based on the Strominger-Yau-Zaslow picture of mirror symmetry. It has been applied successfully to log Calabi-Yau surfaces in the work of \textit{M. Gross} et al. [Publ. Math., Inst. Hautes Étud. Sci. 122, 65--168 (2015; Zbl 1351.14024)]. The starting point for their construction is the data of a \textit{Looijenga pair}, defined as a pair \((Y,D)\) where \(Y\) is a smooth projective surface and \(D\) is a singular reduced normal crossings anticanonical divisor on \(Y\). If \((Y,D)\) is a Looijenga pair satisfying an additional positivity assumption, Gross, Hacking, and Keel construct a family \(\mathcal{X}\rightarrow S\) where \(\mathcal{X}\) is an affine Poisson variety mirror to \(U=Y\setminus D\).
The construction of the mirror variety \(\mathcal{X}\) is based on the consideration of an algebraic object called a \textit{scattering diagram}. The scattering diagram encodes the genus zero log Gromov-Witten invariants of \((Y,D)\), and it is required to satisfy a nontrivial consistency condition. Once the scattering diagram has been constructed and its consistency established, one can construct the algebra \(H^0(\mathcal{X},\mathcal{O}_{\mathcal{X}})\) equipped with a canonical vector space basis whose elements are called \textit{theta functions}.
The paper under review is concerned with quantization of the mirror \(\mathcal{X}\). The goal of quantization is to define a noncommutative \(H^0(S,\mathcal{O}_S)\llbracket\hbar\rrbracket\)-algebra structure on the module \(H^0(\mathcal{X},\mathcal{O}_{\mathcal{X}})\otimes\mathbb{C}\llbracket\hbar\rrbracket\) whose commutator is given at the linear term in \(\hbar\) by the Poisson bracket on \(H^0(\mathcal{X},\mathcal{O}_{\mathcal{X}})\). To define this noncommutative algebra structure, the author first constructs a quantum version of the scattering diagram defined in terms of higher-genus log Gromov-Witten invariants. This quantum scattering diagram is shown to be consistent, and as a consequence, the author obtains the desired noncommutative algebra equipped with a canonical basis of quantum theta functions.
The quantization of the mirror variety \(\mathcal{X}\) has been considered previously by other authors from other points of view. In Section 6 of the paper under review, the author explains how to recover a well known description of the quantized cluster Poisson variety associated to the \(A_2\) quiver. In Section 7, the author explains how the main result can be understood from the perspective of string theory. mirror symmetry; deformation quantization; Gromov-Witten invariants Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Mirror symmetry (algebro-geometric aspects), Relationships between surfaces, higher-dimensional varieties, and physics, Deformation quantization, star products Quantum mirrors of log Calabi-Yau surfaces and higher-genus curve counting | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author considers the natural morphisms from orthogonal and symplectic matrix invariants to the variety of \(\text{GL}_n\)-matrix invariants and studies the relationship between these matrix invariants described by the non-generic fibers of the corresponding morphisms. The main result gives explicit formulas for the number of irreducible components of the reduced variety of a given fiber and the dimensions of these irreducible components. The proofs are based on the recent joint work of the author [\textit{L. Le Bruyn, G. Seelinger}, J. Algebra 214, No. 1, 222-234 (1999; Zbl 0932.16025)]. Recall that \textit{A. Berele, D. J. Saltman} [Isr. J. Math. 63, No. 1, 98-118 (1988; Zbl 0662.16014)] established that the generic fiber of the morphisms under consideration is birationally equivalent to the Brauer-Severi variety of a central simple algebra constructed from the universal division algebra which gives a generic description of the relationship between the corresponding invariants. matrix invariants; symplectic invariants; orthogonal invariants; irreducible components; reduced varieties; generic fibers; Brauer-Severi varieties Trace rings and invariant theory (associative rings and algebras), Actions of groups and semigroups; invariant theory (associative rings and algebras), Actions of groups on commutative rings; invariant theory, Geometric invariant theory, Vector and tensor algebra, theory of invariants Fibers of orthogonal and symplectic matrix 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 author studies the topology of the quotient variety of a complex algebraic projective variety \(X\) with an action of a complex algebraic torus \((\mathbb{C}^*)^ n\). As a main result, he obtains an inductive formula for the intersection Betti numbers. The formula includes singular quotients. One can always find a rationally nonsingular quotient and a canonical map that is a small resolution in the sense of Goresky- MacPherson. The most important quotients under consideration are those that can be understood as symplectic reduced spaces.
The author starts with a nice introduction to this subject. The main part of the paper contains an adequate stratification, the proof that there exist small resolutions and the decomposition theorem for the intersection homology in the symplectic case. In a further step this is done analogously in the so-called semigeometric case which generalizes the symplectic one. But the results are smaller: Vanishing property of intersection homology in odd degree and isomorphism to rational intersection groups are transferable from the fixed-point set to the quotient. Finally, an application to flag varieties is given. See also \textit{F. C. Kirwan}, ``Cohomology of quotients in symplectic and algebraic geometry'', Math. Notes 31 (1984; Zbl 0553.14020).
[See also erratum to this paper in the following review.]. topology of the quotient variety; action of a complex algebraic torus; intersection Betti numbers; symplectic reduced spaces; intersection homology Y. Hu, The geometry and topology of quotient varieties of torus actions, Duke Math. J. 68 (1992) 151--184. Erratum: 68 (1992) 609. Group actions on varieties or schemes (quotients), Homogeneous spaces and generalizations, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) The geometry and topology of quotient varieties of torus 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 is a lecture note on singularity theory. It is shown how to associate to a triple of positive integers \((p_1,p_2,p_3)\) a two-dimensional isolated graded singularity by an elementary procedure that works over any field \(k\) (with no assumptions on characteristic, algebraic closedness or cardinality). This assignment starts from the triangle singularity \(x_1^{p_1} + x_2^{p_2} + x_3^{p_3}\) and when applied to the Platonic (or Dynkin) triples, it produces the famous list of A-D-E-singularities. As another particular case, the procedure yields Arnold's famous strange duality list consisting of the 14 exceptional unimodular singularities (and an infinite sequence of further singularities not treated here in detail). It is shown that weighted projective lines and various triangulated categories attached to them play a key role in the study of the triangle and associated singularities. weighted projective line; (extended) canonical algebra; simple singularity; Arnold's strange duality; stable category of vector bundles Lenzing, H., Rings of singularities, Bull. Iranian Math. Soc., 37, 2, 235-271, (2011) Singularities of surfaces or higher-dimensional varieties, Representations of quivers and partially ordered sets, Cohen-Macaulay modules in associative algebras Rings 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 \(\mathfrak{g}\) be a finite-dimensional complex reductive Lie algebra and \(\text S(\mathfrak{g})\) its symmetric algebra. The nilpotent bicone of \(\mathfrak{g}\) is the subset of elements \((x, y)\) of \(\mathfrak{g} \times \mathfrak{g}\) whose subspace generated by \(x\) and \(y\) is contained in the nilpotent cone. The nilpotent bicone is naturally endowed with a scheme structure, as nullvariety of the augmentation ideal of the subalgebra of \({\text{S}}{\left( \mathfrak{g} \right)} \otimes_{\mathbb{C}} {\text{S}}{\left( \mathfrak{g} \right)}\) generated by the 2-order polarizations of invariants of \({\text{S}}{\left( \mathfrak{g} \right)}\). The main result of this paper is that the nilpotent bicone is a complete intersection of dimension \(3{\left( {{\text{b}}_{\mathfrak{g}} - {\text{rk}}\,\mathfrak{g}} \right)}\), where \({\text{b}}_{\mathfrak{g}}\) and \({\text{rk}}\,\mathfrak{g}\) are the dimensions of Borel subalgebras and the rank of \(\mathfrak{g}\), respectively. This affirmatively answers a conjecture of \textit{H. Kraft} and \textit{N. Wallach} concerning the nullcone [Prog. Math. 278, 153--167 (2010; Zbl 1223.20040)]. In addition, we introduce and study in this paper the characteristic submodule of \(\mathfrak{g}\). The properties of the nilpotent bicone and the characteristic submodule are known to be very important for the understanding of the commuting variety and its ideal of definition. The main difficulty encountered for this work is that the nilpotent bicone is not reduced. To deal with this problem, we introduce an auxiliary reduced variety, the principal bicone. The nilpotent bicone, as well as the principal bicone, are linked to jet schemes. We study their dimensions using arguments from motivic integration. Namely, we follow methods developed by \textit{M. Mustaţă} in [Invent. Math. 145, No. 3, 397--424 (2001; Zbl 1091.14004)]. Finally, we give applications of our results to invariant theory. Charbonnel, J-Y; Moreau, A, Nilpotent bicone and characteristic submodule in a reductive Lie algebra, Transform. Groups, 14, 319-360, (2009) Coadjoint orbits; nilpotent varieties, Arcs and motivic integration, Simple, semisimple, reductive (super)algebras Nilpotent bicone and characteristic submodule of a reductive Lie algebra | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be either the alternating group in 4 elements, \(A_4\), or the Klein four group, \(\mathbb Z_2\times \mathbb Z_2\). Consider the Hurwitz locus \(H_G\subseteq M_g\) of genus \(g\) curves admitting a degree 4 map to \(\mathbb P^1\) with monodromy contained in \(G\). By viewing \(H_G\) as the moduli stack \(M_{0,n}(BG)\) of twisted maps to \(BG\), there is a natural compactification \(\overline H_G\subseteq \overline M_g\) of \(H_G\) given by taking twisted stable maps \(\overline M_{0,n}(BG)\). The main result of the present paper is a computation of generation functions for Hurwitz-Hodge integrals of Hodge classes on \(\overline M_g\) over the \(g\)-dimensional locus \(\overline H_G\). These functions are written as trigonometric expressions in terms of the positive roots of the \(E_6\) and \(D_4\) root systems, which are associated via the McKay correspondence to the subgroups of \(SU(2)\) that are preimage of \(G\hookrightarrow SO(3)\) under the map \(SU(2)\to SO(3)\).
Finally, the authors then prove that the integrals computed in the paper can be interpreted as genus zero Gromov-Witten potentials for the orbifold \(\left [\mathbb C^3/G\right ]\). This orbifold has a Calabi-Yau resolution \(G\)-Hilb\((\mathbb C^3)\to[\mathbb C^3/G]\) given by Nakamura's Hilbert scheme of clusters (cf. \textit{T. Bridgeland, A. King} and \textit{M. Reid} [J. Am. Math. Soc. 14 , No. 3, 535--554 (2001; Zbl 0966.14028)]). The results of the present paper together with the calculation of the Gromov-Witten invariants for \(G-Hilb(\mathbb C^3)\) done by the authors in [Invent. Math. 178, No. 3, 655--681 (2009; Zbl 1180.14010)] for all finite subgroups \(G\subseteq SO(3)\), prove that the crepant resolution conjecture as stated by \textit{J. Bryan} and \textit{T. Graber} [Proc. Symp. Pure Math. 80, Pt. 1, 23--42 (2009; Zbl 1198.14053)] holds for \(A_4\) and for \(\mathbb Z_2\times\mathbb Z_2\). Gromov-Witten; root systems; Hodge integrals; crepant resolution conjecture; orbifolds Bryan, J.; Gholampour, A., Hurwitz-Hodge integrals, the \(E_6\) and \(D_4\) root systems, and the crepant resolution conjecture, Adv. Math., 221, 4, 1047-1068, (2009) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Global theory and resolution of singularities (algebro-geometric aspects) Hurwitz-Hodge integrals, the \(E_6\) and \(D_4\) root systems, and the crepant resolution conjecture | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(A\) be a two-dimensional complete regular local ring: A one-dimensional reduced complete Noetherian local ring \(R=A/I\) is called plane curve singularity; a branch is a height one prime ideal \(p\) of \(A\); if \(p\) and \(q\) are branches, denote by \(\mu (p,q)\) the intersection multiplicity \(l_A(A/(p+q))\) of \(p\) and \(q\). An equivalence relation \(\approx\) on the set of branches of \(A\) is said to be \(\mu\)-compatible if it has the following property: Let \(\{p_1, \cdots , p_r \}\) and \(\{q_1, \cdots, q_r \}\) be sets of branches in \(A\) such that \(p_i \approx q_i\) and \(\mu(p_i,p_j)= \mu(q_i,q_j)\) for all \(i\) and \(j\); then for every branch \(p\) there is a branch \(q \approx p\) such that \(\mu (p,p_j)= \mu(q,q_j)\) for all \(j\). The notion of (a)-equivalence of plane curve singularities introduced by Zariski has been extended by Granja and Sanchez-Giralda to branches of any ring \(A\) as above.
The main purpose of this paper is to give an axiomatic description of (a)-equivalence of branches in \(A\) provided that the residue class field \(k\) of \(A\) satisfies the condition (d): if \(k \subseteq k_0 \subseteq k_1 \subseteq \cdots \subseteq k_s\) is a tower of simple algebraic extensions of \(k\), the set \(\{[l:k_s] |k_s \subseteq l\) simple algebraic\} depends only on the degree sequence \(\{[k_i:k_{i-1}]\mid i=1, \dots , s \}\);
in fact theorem 1 says that if \(k\) satisfies condition (d), (a)-equivalence is the coarsest \(\mu\)-compatible equivalence relation \(\sim\) on the set of branches in \(A\), i.e., if \(\approx\) is any other relation, then \(p \approx q\) implies \(p \sim q\). The author gives also some application of this result. equivalent branches of plane curves Singularities of curves, local rings On the equivalence of plane curve singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The quotient \(M =\Gamma\setminus D\) of a Hermitian symmetric domain by an arithmetic group of automorphisms is a quasiprojective algebraic variety. Varieties like this, variously called modular or locally symmetric or arithmetic varieties, play an important role in representation theory and arithmetic. Many naturally occurring arithmetic varieties are noncompact, and the study of their compactifications has a long history. The variety \(M\) has a canonical embedding in a projective space given by certain automorphic forms for \(\Gamma\) (essentially sections of a power of the canonical bundle); the closure \(M\) in this embedding is the minimal compactification of Satake and Baily-Borel. It is a normal variety which usually has complicated singularities at the boundary and any smooth compactification in which \(M\) is the complement of a normal-crossings divisor dominates it. There is no canonical smooth compactification of \(M\) in general, but \textit{A. Ash}, \textit{D. Mumford}, \textit{M. Rapoport}, and \textit{Y. Tai} [``Smooth compactification of locally symmetric varieties'' (1975; Zbl 0334.14007)] showed how to desingularize \(M^*\) using an extra choice. Given a suitable \(\Gamma\)-admissible rational polyhedral cone decomposition \(\Sigma\) (the notion is recalled in \S\S2, 4), the method produces a smooth projective toroidal compactification \(M^\Sigma\) in which the complement of \(M\) is a divisor with simple normal crossings. There is a morphism
\[
\pi : M^\Sigma\to M^*
\]
extending the identity on \(M\). It is of interest in various questions to study the fibres
of \(\pi\). In this paper I want to describe a homological property of these fibres when the \(Q\)-rank of \(M\) is one (so that \(M^*-M\) is the quotient of a smooth variety by a finite group). I shall assume always that \(\Gamma\) is neat (which can always be achieved by passing to a subgroup of finite index), so that \(M^*- M\) is smooth. In this introduction it will be further assumed that \(M^*-M\) consists of points (i.e. \(D\) is a \(Q\)-rank one tube domain with cusps). Examples include Hilbert modular varieties and arithmetic varieties associated to \(\mathbb Q\)-rank one forms of \(\text{SO}(2,n)\). Modular and Shimura varieties, Global theory and resolution of singularities (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies On the fibres of a toroidal resolution | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author builds ``Tsuchihashi cusps'' [\textit{H. Tsuchihashi}, Tôhoku Math. J., II. Ser. 35, 607-639 (1983; Zbl 0585.14004)] (this is a generalization of Hilbert modular cusp singularities). Such a singularity is defined by a pair \((C,\Gamma)\) of an open convex cone \(C\subset\mathbb{R}^ n\) and a discrete group \(\Gamma\subset GL(n,\mathbb{Z})\) with good conditions. The author defines and studies the notion of ``semi-integral stellable polyhedral cones'' \(C\), the group \(\Gamma\) generated by the reflections with respect to the facets of such a \(C\) gives rise to a good pair \((C,\Gamma)\). There is a duality among stellable cones, the corresponding singularities are dual in the sense of Tsuchihashi [loc. cit.].
At the end, the author gives effective examples of his singularities and computes the arithmetic genus default \(\chi_ \infty\) and the Ogata zeta zero \(Z(0)\) and verifies on his examples the Ogata-Satake conjecture: the \(\chi_ \infty\) of a cusp is equal to the \(Z(0)\) of its dual. A proof of this conjecture is announced as forthcoming.
[See also: \textit{E. B. Vinberg}, Math. USSR, Izv. 5(1971), 1083-1119 (1972); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 35, 1072-1112 (1971; Zbl 0247.20054) and \textit{T. Satake} and \textit{S. Ogata}, in Automorphic forms and geometry of arithmetic varieties, Adv. Stud. Pure Math. 15, 1-27 (1989; Zbl 0712.14009)]. Tsuchihashi cusps; arithmetic genus default; zeta zero Ishida, M.-N.: Cusp singularities given by stellable cones. Int. J. Math.2, 635--657 (1991) Singularities in algebraic geometry, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Global theory and resolution of singularities (algebro-geometric aspects) Cusp singularities given by reflections of stellable cones | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(C\) be a hyperelliptic curve of genus \(g\geq 3\) and \(\mathcal{SU}_C(r)\) the (coarse) moduli space of semistable vector bundles of rank \(r\) with trivial determinant on \(C,\) and \(D\) an effective divisor of degree \(g\) on \(C\). Consider the theta map \[ \theta : \mathcal{SU}_C(r) \dashrightarrow |r \Theta | . \]
In the present paper the authors give a geometric description of the theta map for \(r=2.\) More precisely, they prove that there exists a fibration \[p_D:\mathcal{SU}_C(2) \dashrightarrow |2 \Theta |\cong \mathbb{P}^g\] whose general fiber is birational to \(\mathcal{M}_{0,2g}^{GIT} \) and the theta map restricted to each of these fibers is a \(2\)-to-\(1\) osculating projection up to composition with a birational map. Moreover, they prove that the ramification locus of the theta map has an irreducible component birational to a fibration in Kummer varieties of dimention \(g-1\) over \(|2D|\cong \mathbb{P}^g.\) theta map; osculating projections; moduli spaces; semistable vector bundles Vector bundles on curves and their moduli, Jacobians, Prym varieties, Secant varieties, tensor rank, varieties of sums of powers, Geometric invariant theory, Families, moduli of curves (algebraic) The hyperelliptic theta map and osculating projections | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 Lie algebra over an algebraically closed field \(k\) and \({\mathcal N}\) denote the associated nilpotent cone of \({\mathfrak g}\). Beginning with the investigation of commuting pairs of matrices, the nilpotent commuting variety \({\mathcal C}({\mathcal N}) := \{(x,y) \in {\mathcal N}\times{\mathcal N}~|~ [x,y] = 0\}\) has been long studied. For example, for the Lie algebra of a reductive algebraic group in zero or good characteristic, \textit{A. Premet} [Invent. Math. 154, No. 3, 653--683 (2003; Zbl 1068.17006)] showed that \({\mathcal C}({\mathcal N})\) is equidimensional and characterized its irreducible components.
In this paper, the authors consider the Witt Lie algebra \({\mathfrak g} := W_1\) under the assumption that \(p > 3\). Here \({\mathfrak g}\) admits the structure of a \(p\)-restricted Lie algebra and the nilpotent cone \({\mathcal N}\) happens to agree with the restricted nilpotent cone \({\mathcal N}_p := \{x \in {\mathfrak g}~|~ x^{[p]} = 0\}\). The authors show that \({\mathcal C}({\mathcal N})\) is reducible, equidimensional of dimension \(p\), and not normal. The irreducible components are explicitly described. A key ingredient in the proof is an explicit description of the centralizer of an arbitrary element of \({\mathfrak g}\). To obtain this latter description, the authors make use of the classification of the nilpotent orbits in \({\mathfrak g}\) under the action of its automorphism group that was obtained in previous work of the the first author with \textit{B. Shu} [Commun. Algebra 39, No. 9, 3232--3241 (2011; Zbl 1256.17002)].
The authors also consider the nilpotent commuting varieties for Borel subalgebras. The nilpotent cone for the negative Borel is only one dimensional, so the commuting variety is easily deduced. On the other hand, the problem is more complex for the positive Borel \({\mathfrak b}^+\). Similar to the full Lie algebra, the authors again show that \({\mathcal C}({\mathcal N}({\mathfrak b}^+))\) is reducible, equidimensional of dimension \(p\), and not normal. The irreducible components correspond to all but one of those from \({\mathcal C}({\mathcal N})\). Lastly, the authors observe that this may be applied to describe the spectrum of the cohomology ring of the second Frobenius kernel of the automorphism group of \({\mathfrak g}\). Witt algebra; Cartan type Lie algebra; nilpotent cone; restricted nilpotent cone; nilpotent commuting varieties; Borel subalgebra; cohomology of second Frobenius kernels Yao, Y-F; Chang, H., The nilpotent commuting variety of the Witt algebra, J Pure Appl Algebra, 218, 1783-1791, (2014) Coadjoint orbits; nilpotent varieties, Special varieties, Modular Lie (super)algebras, Cohomology theory for linear algebraic groups The nilpotent commuting variety of the Witt 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 We establish a correspondence between the disk invariants of a smooth toric Calabi-Yau \(3\)-fold \(X\) with boundary condition specified by a framed Aganagic-Vafa outer brane \((L,f)\) and the genus-zero closed Gromov-Witten invariants of a smooth toric Calabi-Yau \(4\)-fold \(\widetilde{X}\), proving the open/closed correspondence proposed by Mayr and developed by Lerche-Mayr. Our correspondence is the composition of two intermediate steps:
\begin{itemize}
\item First, a correspondence between the disk invariants of \((X,L,f)\) and the genus-zero maximally-tangent relative Gromov-Witten invariants of a relative Calabi-Yau \(3\)-fold \((Y,D)\), where \(Y\) is a toric partial compactification of \(X\) by adding a smooth toric divisor \(D\). This correspondence can be obtained as a consequence of the topological vertex (Li-Liu-Liu-Zhou) and Fang-Liu where the all-genus open Gromov-Witten invariants of \((X,L,f)\) are identified with the formal relative Gromov-Witten invariants of the formal completion of \((Y,D)\) along the toric \(1\)-skeleton. Here, we present a proof without resorting to formal geometry.
\item Second, a correspondence in genus zero between the maximally-tangent relative Gromov-Witten invariants of \((Y, D)\) and the closed Gromov-Witten invariants of the toric Calabi-Yau \(4\)-fold \(\widetilde{X}=\mathcal{O}_Y(-D)\). This can be viewed as an instantiation of the log-local principle of van Garrel-Graber-Ruddat in the non-compact setting.
\end{itemize} Gromov-Witten invariants; Calabi-Yau manifolds; toric varieties Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds Open/closed correspondence via relative/local 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 Consider a \(d\)-dimensional reflexive polytope \(\Delta\), its nef-partition (i. e. \(\Delta=\Delta^{(1)}+\ldots+\Delta^{(r)}\) such that for each \(i=1,\ldots,d\) the polytope \(\Delta^{(i)}\) contains the origin and there is a convex integral PL function \(\psi_i\) which has values 1 on every nonzero vertex of \(\Delta^{(i)}\) and 0 on the other polytopes \(\Delta^{(j)}\)), its dual \(\nabla=\nabla^{(1)}+\ldots+\nabla^{(r)}\), and functions \(\omega : \nabla^\vee_{\mathbb Z}\to \mathbb R\), \(\nu : \Delta ^\vee_{\mathbb Z}\to \mathbb R\) (where \(\nabla^\vee=\text{conv}\{\Delta^{(1)},\ldots, \Delta^{(r)}\}\), \(\Delta^\vee=\text{conv}\{\nabla^{(1)},\ldots, \nabla^{(r)}\}\)). Based on this data, the authors construct a pair of affine structures on a sphere (or a product of spheres) with a codimension 2 discriminant locus and whose monodromy representations have dual linear parts.
If the functions \(\omega\) and \(\nu\) are integer-valued, then one can associate to this data a one-parameter family of complete intersections in a toric variety. The authors do not study the geometry of this degeneration, but clarify the combinatorics of the model. In particular, they establish a homeomorphism between it and a sphere, which positively solves the conjecture about sphericity of the Clemens complex of a maximally degenerate complete intersection family. sphere; toric variety; reflexive polytope; monodromy C. Haase and I. Zharkov, ''Integral affine structures on spheres: complete intersections,'' Int. Math. Res. Not., vol. 2005, iss. 51, pp. 3153-3167, 2005. Toric varieties, Newton polyhedra, Okounkov bodies, Real algebraic and real-analytic geometry, Calabi-Yau manifolds (algebro-geometric aspects) Integral affine structures on spheres: complete intersections | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this paper, there is a new proof of the existence and a construction of a resolution of excellent schemes of finite type over a ground field of characteristic 0 and of principalization of an ideal of a smooth scheme.
This proof is based on the earlier works of Villamayor, Villamayor-Encinas and Bierstone-Milman, all working after Hironaka. The main point is as usual the ``maximal contact''.
What is new~?
\noindent 1. This proof is concise (14 pages).
2. The Hilbert-Samuel function is avoided.
The proof starts as usual by the construction of a ``basic object'' \((W_0, (J_0,b),E)\) where \(W_0\) is a smooth variety, \(J_0\subset{\mathcal O}_{W_0}\) an ideal, \(b\in \mathbb N\), \(E_0\) is a normal crossing divisor of \(W_0\). The proof is in two parts:
\noindent 1- Resolve any basic object,
\noindent 2- Prove that the resolution of basic objects leads to resolution of embedded schemes and to principalization of ideals.
In the case of desingularization, the main invariant used is the order of \(J_0\) restricted to \(W_0\) a smooth variety of maximal contact. The strata given by this order are not the Hilbert-Samuel strata. This is an improvement: these new strata can be computed easily, which is not the case for the H-S strata. Hence, this new algorithm is easier to implement.
A problem arises: the authors do not look at the strict transform of \({J_0}\), but at its weak transform \(t^{-b}{J_0}\) where div\((t)\) is the exceptional divisor of the blowing-up. This small modification with the usual proofs simplifies the redaction but, then, it is not clear at all that the algorithm is independent of the embedding: the weak transform \(t^{-b}{J_0}\) depends obviously on the embedding and on the choice of \(W_0\). This difficulty is easily overcome: the authors show that two different \(W_0\) and \(W'_0\) have same dimension and that you can find an étale covering \(\tilde W_0\) where the pull back of the basic objects on \(W_0\) and \(W'_0\) are the same, so the algorithms coincide on each pair \((W_0, \tilde W_0)\) and \((W'_0, \tilde W_0)\). This means that the authors' construction has many properties of invariance that should be interpreted geometrically. Encinas S., Villamayor O.: A proof of desingularization over fields of characteristic zero. Rev. Mat. Iberoamericana 19(2), 339--353 (2003) Global theory and resolution of singularities (algebro-geometric aspects), Modifications; resolution of singularities (complex-analytic aspects) A new proof of desingularization over fields of 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 In this paper, the main result says that if \(X\) and \(X'\) are two schemes satisfying the mild condition (ELF), defined below, and if the formal completions of \(X\) and \(X'\) along their singularities are isomorphic, then the idempotent completions of the triangulated categories of singularties are equivalent. Here the condition (ELF) means that the scheme is separated, Noetherian of finite Krull dimension, and for any coherent sheaf \(\mathcal{F}\) there exists a locally free sheaf \(\mathcal{E}\) and an epimorphism \(\mathcal{E} \rightarrow \mathcal{F}\). The triangulated category of singularities \(\mathbb{D}_{\text{Sg}}(X)\) of \(X\) is the quotient of the bounded derived category of the abelian category of coherent sheaves on the scheme, by the full triangulated subcategory of perfect complexes.
Then the author proves that for a morphism \(f : X \rightarrow X'\) which is an isomorphism infinitely near a closed subscheme \(Z \subset X\) containing the singular locus of both schemes satisfying (ELF) \(X\) and \(X'\), then the induced functor \(\bar{f}^*\) is fully faithful and, moreover, any object \(B \in \mathbb{D}_{\text{Sg}}(X')\) is a direct summand of some object coming from \(\mathbb{D}_{\text{Sg}}(X)\).
Let \(\mathfrak{Perf}(X)\) be the triangulated category of perfect complexes. \(\mathfrak{Perf}_{\text{sing} X}(X)\) be \(\mathfrak{Perf}(X) \cap \mathbb{D}^b_{\text{sing} X}(\text{coh} X)\) of perfect complexes supported in \(\text{sing} X\). In the paper the author proves and uses a proposition that the idempotent completions of \(\mathbb{D}_{\text{Sg}}(X)\) and \(\mathbb{D}^b_{\text{sing} X}(\text{coh} X) / \mathfrak{Perf}_{\text{sing} X}(X)\) are equivalent. In the last section, he proves that there is a difference between \(K_{-1}\) of \(\mathfrak{Perf}(X)\) and \(\mathfrak{Perf}_{\text{sing} X}(X)\). He also gives an example where the triangulated category of singularities of two schemes constructed differently, are equivalent. triangulated categories of singularities; idempotent completions A. Căldăraru and J. Tu, Curved \(A\)\_{}\{\(\infty\)\} algebras and Landau-Ginzburg models, arXiv:1007.2679. Derived categories, triangulated categories, Singularities in algebraic geometry Formal completions and idempotent completions of triangulated categories 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 (V,p) be a normal surface singularity, and \(\pi : M\to V\) be the minimal resolution. We assume that (V,p) is rational, i.e. \(R^ 1\pi_*{\mathcal O}_ M=0\). Let \(\omega : {\mathcal M}\to R\) be a 1-convex flat representative of the semi-universal deformation of the germ of M at the exceptional set, where R is a complex manifold. Then \(\omega\) can be simultaneously blown down to a deformation of the singularity (V,p). There is an induced holomorphic map germ \(\phi : (R,0)\to (S,0)\) to the base space of the semi-universal deformation \(\vartheta : ({\mathcal V},p)\to (S,0)\) of (V,p). It was proved by M. Artin that \(\phi\) is finite onto its image, and that the germ of its image \(S_ a=\phi (R)\) is an irreducible component of the base space (S,0). This component is called the Artin component. - If (V,p) is a rational double point (RDP), then \((S_ a,0)\) is the whole base space (S,0). It was shown by E. Brieskorn that in this case the map germ \(\phi\) can be represented by a Galois covering with group W. Here the weighted dual graph associated to the minimal resolution of such a singularity is a Dynkin diagram of type \(A_ k, D_ k, E_ 6, E_ 7\) or \(E_ 8\), and W is the corresponding Weyl group. Furthermore the discriminant \(\Delta\) \(\subset S\) of the semi- universal deformation \(\vartheta\) is an irreducible hypersurface such that the fiber over a generic point of \(\Delta\) has only one singularity which is an ordinary double point.
The main results of the paper are generalizations of these results to arbitrary rational singularities. The author shows that in the general case \(\phi\) can be represented by a Galois covering with group W which is isomorphic to a direct product of Weyl groups corresponding to the maximal RDP-configurations on the minimal resolution of the given rational singularity. He describes the components of the discriminant \(\Delta_ a=\Delta \cap S_ a\) of the Artin component and the singularities of the fibers corresponding to generic points of \(\Delta_ a.\)
In the formal category the same results are valid and were obtained earlier under an additional assumption by \textit{J. Wahl} [Compos. Math. 38, 43-54 (1979; Zbl 0412.14008) and Duke Math. J. 46, 341-375 (1979; Zbl 0472.14002)]. simultaneous resolution; discriminant of semi-universal; deformation; normal surface singularity; Artin component; rational singularities; Weyl groups Deformations of singularities, Singularities of surfaces or higher-dimensional varieties, Deformations of complex singularities; vanishing cycles, Singularities in algebraic geometry On the discriminant of the Artin component | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The philosophy in the article is that any moduli space of algebras of a fixed dimension, can be naturally stratified by orbifolds. The meaning of this is of course not very stringent as we don't know if there exists a moduli space, and the stratification in orbifolds could be given in several ways. However, assuming that a moduli does exist, and letting the orbifolds be equivariant versal families, the deformation theory and cohomology in particular cases, gives possible constructions of, and knowledge about moduli.
In the case of complex Lie algebras, the orbifolds consist of quasi-projective spaces. They are obtained by removing a divisor from \(\mathbb P^n_{\mathbb C}\) together with an action of a symmetric group.
The main result of the article is that the strata that consists of more that one point can be given projective coordinates. The stratification is given in a unique form by applying deformation theory: The strata are given as smooth deformations, and the strata are connected by jump deformations. The authors also state that deformations are smooth along the families of points to which they jump, which is a very good property of the stratification.
The authors find that each family (with more than one point) of solvable complex Lie algebras of dimension \(d\), \(d\leq 5\), contains one special nilpotent element. Then each family can be given by the action of a symmetric group on \(\mathbb P^n_{\mathbb C}\), and the generic point of \(\mathbb P^n_{\mathbb C}\) corresponds to the nilpotent. The meaning of this is that if \(\mathbb P^n_{\mathbb C}\) is given by projective coordinates \((p_0,\dots,p_n)\), then the generic point is \((0,\dots,0)\). The authors claim that this point is usually excluded by algebraic geometers, but it has to be present in this picture as there is an algebra corresponding to the generic point. Thus we can read out of the article that by geometric invariant theory there is a dense orbit, so that there does not exist an algebraic quotient. The generic element in one family may be isomorphic to the generic point in another family, and this represents the only overlap between the families.
The authors point out that, contrary to earlier results by them and others, the goal of this article is not to simply determine a list of the algebras, but also to understand how the moduli space is glued together. Then the cohomology of the algebras has to be computed, and in addition the versal deformations of the algebras. The result is a more natural decomposition of the moduli space, and divides it into fewer strata than earlier results.
The article starts by applying this new view to make a summary of the moduli spaces of \(3\) and \(4\)-dimensional algebras. The lists are given with the dimension of the cohomology of interest, and the versal deformations are included to some extent. Then a more thorough discussion of the \(5\)-dimensional complex Lie algebras is given:
Given an exact sequence \(0\rightarrow M\rightarrow L\rightarrow W\rightarrow 0\) where \(L\) is a Lie-algebra which is an extension of \(W\) by \(M\). \(M\) is an ideal in \(L\), and \(W\) is the quotient algebra \(W=L/M\). The algebra structures of \(M\), \(L\), and \(W\) is denoted \(\mu\), \(d\), \(\delta\) respectively. One can write \(d=\delta+\mu+\lambda+\psi\) where \(\lambda\) and \(\mu\) are additional algebra structures. Then \(d\) is a algebra structure exactly when the \textit{compatibility condition}, the \textit{Maurer-Cartan condition}, and the \textit{cocycle condition} are satisfied. This is the basis of the classification of the algebras, and the computation of the versal deformation which is the main result of the article. The article is easy to read, it classifies the complex Lie algebras of dimension five, and gives the essential information about the moduli space. The article serves as a main step to the general classification of finite dimensional complex Lie algebras. complex finite dimensional Lie algebras; versal deformation; jump deformation; orbifold; Maurer-Cartan condition; compability condition; cocycle condition; solvable Lie algebra; nilpotent Lie algebras; stratification of moduli; moduli of algebras Fialowski, A.; Penkava, M., The moduli space of complex \(5\)-dimensional Lie algebras, J. Algebra, 458, 422-444, (2016) Formal methods and deformations in algebraic geometry, Deformations and infinitesimal methods in commutative ring theory, Local deformation theory, Artin approximation, etc., Deformations of associative rings, (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.), Homological methods in Lie (super)algebras, Graded Lie (super)algebras The moduli space of complex 5-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 [For the entire collection see Zbl 0624.00008.]
This talk is mainly a report on some joint work with \textit{J. T. Stafford} [Proc. Lond. Math. Soc., III. Ser. 56, No.2, 229-259 (1988; see the preceding review)]. That paper examines the structure of \({\mathcal D}(X)\), the ring of differential oprators on an irreducible affine curve X, defined over an algebraically closed field k of characteristic zero. \({\mathcal D}(X)\) is a finitely generated k-algebra and right and left noetherian. However, in contrast to the non-singular case, \({\mathcal D}(X)\) need not be a simple ring if X is singular. In theorem 2.3 it is seen that the simplicity of \({\mathcal D}(X)\) is equivalent to a number of other properties. In particular, \({\mathcal D}(X)\) is simple if and only if the natural projection \(\pi: \tilde X\to X\) from the normalisation is bijective. When \({\mathcal D}(X)\) is not simple, there is a unique minimal non-zero ideal J(X), and \(H(X):={\mathcal D}(X)/J(X)\) is a finite dimensional k-algebra. The ring of regular functions \({\mathcal O}(X)\) need not be a simple \({\mathcal D}(X)\)-module, but it has a unique simple submodule J(X).\({\mathcal O}(X)\), and \(C(X):={\mathcal O}(X)/J(X).{\mathcal O}(X)\) is a finite dimensional k-algebra. Both H(X) and C(X) split as a direct sum of finite dimensional algebras, \(H_ x\) and \(C_ x\), one for each singular point \(x\in Sing(X).\)
The algebras \(H_ x\) and \(C_ x\) depend only on the local ring \({\mathcal O}_{X,x}\), and {\S} 3 examines how the structure of \(H_ x\) and \(C_ x\) depends on that of \({\mathcal O}_{X,x}\). ring of differential operators on an irreducible; singular curve; ring of regular functions Singularities of curves, local rings, Modules of differentials, Commutative Artinian rings and modules, finite-dimensional algebras Curves, differential operators and finite dimensional algebras | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(\Sigma\) be a compact Riemann surface of genus \(g\geq 1\) and \(G^c\) a complex reductive Lie group. A Higgs bundle for the group \(G^c\) is a pair \((E,\Phi)\) where \(E\) is a holomorphic principal \(G^c\)-bundle and \(\Phi\) is a section of \(\mathrm{ad}(E)\otimes K\) -- the adjoint bundle, twisted by the canonical line bundle \(K\) of \(\Sigma\). It is well known that the moduli space \(\mathcal M_{G^c}\) of Higgs bundles, over \(\Sigma\), for the group \(G^c\), is a completely integrable Hamiltonian system. In other words, there is a proper map
\[
h:\mathcal M_{G^c}\to\mathcal A,
\]
where \(\mathcal A\) is an affine space of half of the dimension of \(\mathcal M_{G^c}\) whose generic fiber is an abelian variety. This is the so-called Hitchin system (and \(h\) is called the Hitchin map). More precisely, since \(\mathrm{ad}(E)\) is a bundle of Lie algebras, isomorphic to \(\mathfrak{g}\), and if \(p\) is an invariant homogeneous polynomial on \(\mathfrak{g}\) of degree \(d\), it makes sense to evaluate \(p\) at \(\Phi\) and \(p(\Phi)\in H^0(\Sigma, K^d)\). So, by taking a basis \(p_1,\ldots,p_k\) for the ring of invariant homogeneous polynomials on \(\mathfrak{g}\), then \(\mathcal A=\bigoplus_{i=1}^kH^0(\Sigma,K^i)\) and \(h(E,\Phi)\) is given by evaluating all these \(p_i\) at \(\Phi\).
For \(G^c=\mathrm{GL}(n,\mathbb{C})\), a Higgs bundle is given by a rank \(n\) vector bundle \(V\) and by a section \(\Phi\) of \(\mathrm{End}(V)\otimes K\). The Hitchin map takes \((V,\Phi)\) to the coefficients of the characteristic polynomial of \(\Phi\) and the generic fiber is the Jacobian of the so-called spectral curve. In a little more detail, given a generic element \(s=(s_1,\ldots,s_n) \in\bigoplus_{i=1}^nH^0(\Sigma,K^i)\) the spectral curve \(S\) corresponding to \(s\) is a curve inside the total space of \(K\), defined by
\[
x^n+\pi^*s_1x^{n-1}+\cdots+\pi^*s_n=0,
\]
where \(\pi:K\to X\) is the projection and \(x\) is the tautological section of \(\pi^*K\). In other words, \(S\) is defined by the equation \(\det(xI-\Phi)=0\) where \(h(V,\Phi)=s\). Then \(h^{-1}(s)\cong\mathrm{Jac}(S)\) basically because any stable \((V,\Phi)\in h^{-1}(s)\) can be uniquely written as \((\pi_*L,\pi_*x)\) with \(L\in\mathrm{Jac}(S)\). If instead one considers \(G^c=\mathrm{SL}(n,\mathbb{C})\), then only line bundles on the Prym variety of \(S\) are allowed so that the determinant of \(V\) is trivial, so the fiber is \(\mathrm{Prym}(S)\). See [\textit{N. Hitchin}, Duke Math. J. 54, 91--114 (1987; Zbl 0627.14024)] for other examples of the Hitchin system for other complex groups.
This abelianization process plays a central role in many important aspects of Higgs bundle theory, not only on the integrable systems side, but also, for example, on the study of special representations of surface groups (known as Hitchin representations) or on the study of the Langlands duality phenomena.
The notion of Higgs bundles generalizes however to any real reductive Lie group, and the present paper addresses the question of describing the Hitchin system for the real Lie groups \(\mathrm{SL}(m, \mathbb{H})\), \(\mathrm{SO}(2m,\mathbb{H})\) and \(\mathrm{Sp}(m,m)\). In doing so, the authors discover that the above-mentioned abelianization process does not hold anymore for these real forms, in the sense that the generic fiber of the Hitchin map is now described, not in terms of line bundles over the spectral curve, but in terms of rank 2 vector bundles over that curve (or quotients of it). More precisely, the authors prove that for \(\mathrm{SL}(m, \mathbb{H})\), the fiber consists of the moduli space of semi-stable rank \(2\) vector bundles with fixed determinant on the spectral curve \(S\); for \(\mathrm{SO}(2m,\mathbb{H})\), the fiber has several components, each of which is a moduli space of semi-stable rank \(2\) bundles on a quotient \(\overline S\) of the spectral curve; for \(\mathrm{Sp}(m,m)\) it is a \(\mathbb{Z}_2\)-quotient of a moduli space of semi-stable rank \(2\) parabolic bundles on \(\overline S\). Hitchin, NJ; Schaposnik, LP, Nonabelianization of Higgs bundles, J. Differ. Geom., 97, 79-89, (2014) Relationships between algebraic curves and integrable systems, \(G\)-structures, Vector bundles on curves and their moduli, Vector bundles on surfaces and higher-dimensional varieties, and their moduli Nonabelianization of 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 For part I of this paper see: http://front.math.ucdavis.edu/math.AG/981130.
From the paper: Let \(E\) be a smooth elliptic curve and let \(G\) be a simple complex algebraic group of rank \(r\). We shall always assume that \(\pi_1(G)\) is cyclic and \(c\) is a generator. The goal of this paper is to continue the study, begun in [part I], of the moduli space \({\mathcal M}(G,c)\) of semistable holomorphic \(G\)-bundles \(\xi\) with \(c_1(\xi)=c\). In part I this space was studied from the transcendental viewpoint of \((0,1)\)-connections using the results of Narasimhan-Seshadri and Ramanathan that in every \(S\)-equivalence class there is a unique representative whose holomorphic structure is given by a flat connection. This viewpoint, however, is not suitable for many questions, such as finding universal bundles, studying singular elliptic curves, or generalizing to families of elliptic curves. In this paper, which is largely independent of [part I], we describe \({\mathcal M}(G,c)\) from an algebraic point of view.
The moduli space is constructed by considering deformations of a minimally unstable \(G\)-bundle. The set of all such deformations can be described as the \(\mathbb{C}^*\)-quotient of the cohomology group of a sheaf of unipotent groups, and we show that this quotient has the structure of a weighted projective space. We identify this weighted projective space with the moduli space of semistable \(G\)-bundles, giving a new proof of a theorem of \textit{E. Looijenga} [Invent. Math. 38, 17--32 (1976; Zbl 0358.17016)]. Friedman, R; Morgan, JW, Holomorphic principal bundles over elliptic curves. II. the parabolic construction, J. Differ. Geom., 56, 301-379, (2000) Families, moduli of curves (algebraic), Algebraic moduli problems, moduli of vector bundles, Vector bundles on curves and their moduli Holomorphic principal bundles over elliptic curves. II: The parabolic construction. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \({\mathcal X}\) be a genus \(g\geq 2\), non-singular, complete algebraic curve defined over an algebraically closed field of characteristic \(p \geq 0\). The automorphism group \(G = \mbox{Aut}({\mathcal X})\) of \({\mathcal X}\) is a finite group which acts naturally on the space \({\Omega_{\ell}}={\Omega_{\ell}}({\mathcal X})\) of holomorphic \(\ell\)-differentials on \({\mathcal X}\) for any \(\ell \geq 1\). Recall that the dimension of \({\Omega_1}\) is \(g\) and the dimension of \({\Omega_2}({\mathcal X})\) is \(3g-3\), for \(g\geq 2\).
A theorem of Max Noether says that: if \({\mathcal X}\) is non-hyperelliptic of genus \(g \geq 3\), then the product map \(\mbox{Sym}^{\ell} ({\Omega_1}) \, \, \to \, \, {\Omega_{\ell}}\) is surjective for \(\ell \geq 2\). Thus for \(\ell=2\) the kernel \(\Lambda_2=\Lambda_2(X)\) of this map has dimension \(\frac {(g-2)(g-3)} 2\). Here \(\mbox{Sym}^{\ell}\) denotes the \(\ell\)-th symmetric power of a vector space (i.e., the space of homogeneous polynomials of degree \(\ell\) in the coordinates of a basis of the dual of the vector space).
{Petri's Theorem:} Assume that \(g\geq 4\) and \({\mathcal X}\) is neither hyperelliptic, trigonal or isomorphic to a smooth quintic plane curve. Then the ideal \(I\) of the algebra \(\mbox{Sym}({\Omega_1} ) \) generated by \(\Lambda_2({\mathcal X})\) is the ideal corresponding to a \(G\)-invariant curve \({\mathcal X}'\) in \({\mathbb P} ({{\Omega_1}}^{\ast})\) that is isomorphic to \({\mathcal X}\) as a \(G\)-curve. Further, \(I\cap \mbox{Sym}^2({\Omega_1})=\Lambda_2({\mathcal X})\).
For curves satisfying Petri's theorem, the authors give a criterion for identifying the automorphism group as an algebraic subgroup the general linear group. Furthermore, the action of the automorphism group is extended to a linear action on the generators of the minimal free resolution of the canonical ring of the curve \({\mathcal X}\). They prove that the automorphism group of a curve \({\mathcal X}\), as a finite set, can be seen as a subset of the \(g^2 (g + 1)^2-1\)-dimensional projective space and can be described by explicit quadratic equations. At the end of the paper they illustrate some of their results with Fermat curves. algebraic curve; canonical ideal; automorphism group Automorphisms of curves, Syzygies, resolutions, complexes and commutative rings Automorphisms and the canonical ideal | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 quotient of a projective homogeneous space \(G/P\), where \(G\) is a semisimple algebraic group defined over a field \(k\) of characteristic zero and \(P\) is a maximal parabolic subgroup of \(G\), by the action of a maximal torus \(T\) of \(G\). The author's goal is to obtain a description of the automorphism group of this quotient and thus classify its \(k\)-forms. More precisely, if \(\rho: G\to \mathrm{GL}(V)\) is an irreducible finite-dimensional representation of \(G\) with highest weight \(\omega_i\), \(v\in V\) is a vector of highest weight, \(P\) is the stabilizer of the line \(kv\in\mathbb{P}(V)\), \(Y\subset V\setminus\{0\}\) is the affine cone over \(G/P\) with removed origin, the author considers an open subset \(U\subset Y\) (defined explicitly as the intersection of \(U\) with the set of \(G\)-stable points in \(V\) having trivial stabilizer) and studies the quotient \(X=U/T\). His approach is based on the theory of \(X\)-torsors under tori.
In the first section, he explains some generalities regarding lifting automorphisms to torsors. This works especially well under the assumption that the complement to \(U\) in \(Y\) has codimension more than one (in this case the natural map \(U\to X\) defines a universal torsor). The cases where this assumption does not hold are listed in Proposition 2.1. Some more cases are excluded to fit into the hypotheses of the Demazure--Tits theorem giving a description of \({\text{{Aut}}}(\overline{G/P})\). Modulo these exceptions, Theorem 2.2 gives an explicit description of \({\text{{Aut}}}(\overline{X})\). This yields a description of \(\bar k/k\)-forms of \(X\). Using results of \textit{P.~Gille} [J. Ramanujan Math. Soc. 19, 213--230 (2004; Zbl 1193.20057)] and \textit{M.~S.~Raghunathan} [J. Ramanujan Math. Soc. 19, 281--287 (2004; Zbl 1080.20042)], the author shows that any such form is a quotient of a homogeneous space of a quasisplit form of \(G\) by a maximal torus (Theorem 2.4). This implies that all such forms are \(k\)-unirational. homogeneous spaces; tori; twisted forms; torsors Automorphisms of surfaces and higher-dimensional varieties, Group actions on varieties or schemes (quotients), Homogeneous spaces and generalizations Automorphisms and forms of toric quotients of homogeneous 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 authors introduce the notion of a quaternionic \(r\)-Kronecker structure of rank \(k\) on a manifold \(M\). This is a certain type of bundle map \(\alpha:E\otimes \mathbb{C}^r\to T^{\mathbb{C}}M\), where \(E\) is a quaternionic vector bundle of rank \(k\). In the case where \(\alpha\) is an isomorphism and \(r=2\), this reduces to an almost hypercomplex structure. They also introduce a complex analog, as well as twistor spaces of (integrable and regular) Kronecker structures. These are complex manifolds admitting a natural holomorphic submersion to \(\mathbb{C}\mathrm{P}^{r-1}\) from which the original manifold may be recovered as a suitable space of sections, analogous to the so-called twistor lines in the hypercomplex case. Conversely, it is proven that for any complex manifold \(Z\) with a surjective holomorphic submersion \(\pi:Z\to \mathbb{C}\mathrm{P}^{r-1}\), certain spaces of sections carry natural \(r\)-Kronecker structures.
After discussing connections to hypercomplex geometry, the authors apply these concepts to certain open subsets \(M_{d,g}\) of the Hilbert scheme of curves of fixed genus \(g\) and degree \(d\) in \(\mathbb{C}\mathrm{P}^3\), showing that the subspace \(M_{d,g}^\sigma\) of real curves admits a quaternionic \(4\)-Kronecker structure of rank \(2d\) (though it may be empty for some values of \(d,g\)). They also consider curves in \(\mathbb{C}\mathrm{P}^n\) for \(n>3\), where the assumptions needed to ensure smoothness impose stronger constraints on \(d\) and \(g\). In the final section, the authors return to the case \(n=3\), and study \(M_{1,0}^\sigma=S^4\) in more detail. projective curves; quaternionic structures; twistor methods Hyper-Kähler and quaternionic Kähler geometry, ``special'' geometry, Twistor methods in differential geometry, Parametrization (Chow and Hilbert schemes), Plane and space curves Differential geometry of Hilbert schemes of curves in a projective space | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Decompositions of simple singularities in terms of their Dynkin diagrams have been described by O. V. Lyashko [Geometry of bifurcation diagrams, J. Sov. Math. 27, 2736-2759 (1984); transl. from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 22, 94-129 (1983; Zbl 0548.58024)]. The ''next'' group of singularities are parabolic, i.e.
\[
P_ 8(\lambda): x^ 3 + y^ 3 + z^ 3 + \lambda xyz,\quad \lambda^ 3 \neq -27;
\]
\[
X_ 9(\lambda): x^ 4 + y^ 4 + \lambda x^ 2 y^ 2,\quad\lambda \neq 4;
\]
\[
J_{10}(\lambda): y^ 3 + x^ 6 + \lambda y^ 2 x^ 2,\quad 4 \lambda^ 3\neq -27.
\]
Decompositions of \(P_ 8\) and \(X_ 9\) have been examined by J. W. Bruce and C. T. C. Wall [J. Lond. Math. Soc. 19, 245-265 (1979; Zbl 0406.14020)] and by J. W. Bruce and P. J. Giblin [Proc. Lond. Math. Soc. 42, 270-298 (1981; Zbl 0403.14004)]. In the reviewed article the author describes decompositions of parabolic singularities \(J_{10}\), namely Theorem: Decompositions on one level of singularity \(J_{10}(\lambda)\) are all subdecompositions of the following: \(E_ 8\), \((E_ 7,A_ 1)\), \((E_ 6,A_ 2)\), \(D_ 8\), \((D_ 6,2A_ 1))\), \((D_ 5,A_ 3)\), \(2D_ 4\), \(A_ 8\), \(A_ 7,A_ 1)\), \((A_ 5,A_ 2,A_ 1)\), \(2A_ 4\), \((2A_ 3,2A_ 1)\), \(4A_ 2\). All of the above take place for every lambda with \(4 \lambda^ 3 = -27\). simple and parabolic singularities Differentiable maps on manifolds, Theory of singularities and catastrophe theory, Singularities of surfaces or higher-dimensional varieties, Singularities of differentiable mappings in differential topology Decompositions of parabolic singularities of one level | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \((X,\Delta)\) a complex projective pair with log canonical singularities, a \textit{lc-trivial fibration} \(f: X \to Y\) is a morphism such that \(K_X+\Delta\) is (\({\mathbb Q}\)-equivalent) to the pull-back by \(f\) of a \({\mathbb Q}\)-Cartier
\({\mathbb Q}\)-divisor \(D\) on \(Y\). A relevant question here is to know if there exists a log canonical structure \((Y, \Delta_Y)\) such that \(D\) is \(K_Y+\Delta_Y\). It is known that \(D\) decomposes as \(K_Y+B_Y+M_Y\) where \(B_Y\) encodes information on the singularities of \(f\) and \(M_Y\), called the moduli divisor, (conjecturally) on the birational variation of the fibres of \(f\). It is an important conjecture on \(M_Y\) (see the B-Semiampleness Conjecture in the Introduction of the paper under review and references therein) that in a particular birational model, the so called Ambro model of \(f\), if \(\Delta\) is effective over the generic point of \(Y\) then the moduli divisor is semiample.
The main result of the paper under review is (see Theorem. A) is that under the assumption of the B-semiamplenes conjecture in \(\dim Y -1\), the moduli part of a lc-trivial fibration is semiample when restricted to any divisorial valuation over its Ambro model. Moreover, (see Theorem. B), under the assumption of \(M_Y\) big and considering only components of the augmented base locus of \(M_Y\), the assumption on the B-semiampleness conjecture can be relaxed up to \(\dim Y-2\). canonical bundle formula; moduli divisor Families, moduli, classification: algebraic theory, Minimal model program (Mori theory, extremal rays), Divisors, linear systems, invertible sheaves, Adjunction problems, Families, moduli of curves (algebraic) On the B-Semiampleness Conjecture | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In 2001 \textit{B. Dubrovin} and \textit{Y. Zhang} [``Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants'', Preprint, \url{arXiv:math/0108160}] gave a complicated explicit formula for the genus \(2\) generating function for semisimple Frobenius manifolds. In 2012, in collaboration with \textit{S.-Q. Liu} [Russ. J. Math. Phys. 19, No. 3, 273--298 (2012; Zbl 1325.53114)], they split it into contributions from genus \(2\) dual graphs and the so called genus \(2\) \(G\)-function \(G^{(2)}\). In 2015 \textit{X. Liu} and \textit{X. Wang} [Adv. Math. 274, 631--650 (2015; Zbl 1368.53058)] showed that \(G^{(2)}\) vanishes for simple singularities and \(\mathbb{P}^1\) orbifolds of Fano type. In this paper the author studies properties of \(G^{(2)}\) for the cubic elliptic singularity and derives closed form formulas for its derivatives along the mirror map.
The derivation is based on the Saito-Givental theory for isolated polynomial singularities. First, Frobenius manifold structure is put on the deformation space of the singularity using Saito's relative differential form, which is semisimple at the generic point. Then Gromov-Witten invariants and the \(I\)-function are defined using Givental's formalism. From the \(I\)-function two series, \(X\) and \(L\), are defined, which are quasi-modular forms with the modular group \(\Gamma_0(3)\) up to analytic continuation and symplectic transformation. Finally, the derivative of \(G^{(2)}\) is expressed as an explicit polynomial in \(X\) and \(L\). The modular properties are naturally interpretable in terms of Gromov-Witten invariants of the corresponding \((3,3,3)\) type elliptic orbifold. Saito-Givental theory; semisimple Frobenius manifold; Givental quantization; genus 2 G-function; cubic elliptic singularity; quasi-modular forms Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds The genus two G-function for the cubic elliptic singularity | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This is a very interesting paper. The authors state in their summary: ``Given an integer \(b\) and a finitely presented group \(G\), we produce a compact symplectic 6-manifold with \(c_1 =0, b_2 >b, b_3 >b\) and \(\pi _1 =G\). In the simply connected case, we can also arrange for \(b_3 =0\); in particular, these examples are not diffeomorphic to Kähler manifolds with \(c_1 =0\) (otherwise, \(b_3 \geq 2\)). The construction begins with a certain orientable, four-dimensional, hyperbolic orbifold assembled from right-angled 120-cells. The twistor space of the hyperbolic orbifold is a symplectic Calabi-Yau orbifold; a crepant resolution of this last orbifold produces a smooth symplectic manifold with the required properties.''
This means that it is different from the Kähler case, in the symplectic case, there are infinitely many topological six manifolds which admit symplectic Calabi-Yau structures. For the Kähler case, see the reviewer's review of [\textit{D. Joyce}, in: Geometry of special holonomy and related topics. Leung, Naichung Conan (ed.) et al., Somerville, MA: International Press. Surveys in Differential Geometry 16, 125--160 (2011; Zbl 1255.14030)].
The authors in this paper continue the works in [\textit{A. G. Reznikov}, Ann. Global Anal. Geom. 11, No. 2, 109--118 (1993; Zbl 0810.53056)]; [\textit{J. Fine} and \textit{D. Panov}, J. Differ. Geom. 82, No. 1, 155--205 (2009; Zbl 1177.32014)] and [Geom. Topol. 14, No. 3, 1723--1763 (2010; Zbl 1214.53058)]. Calabi-Yau symplectic six-fold; hyperbolic fourfold; fundamental groups; Betti numbers Fine, J., Panov, D.: The diversity of symplectic Calabi-Yau six-manifolds preprint (2011). arXiv:1108.5944 Calabi-Yau manifolds (algebro-geometric aspects), Global theory of symplectic and contact manifolds, Hyperbolic groups and nonpositively curved groups The diversity of symplectic Calabi-Yau 6-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 In this survey article, the author explains several connections between the Betti numbers and shifts occurring in a graded minimal free resolution of the homogeneous coordinate ring of a finite set of points in \(\mathbb{P}^n\), and the geometry of the configuration of those points. The ranks of the modules forming the linear part of such a resolution can be computed using the Koszul complex and some easy linear algebra. The author reviews this method in a very elementary and readable style. She shows how it was used in a series of joint papers with \textit{M. E. Rossi} and \textit{G. Valla} to study the geometric meaning of the length of the linear part of that resolution. For instance, she sketches a new proof of the strong Castelnuovo lemma [cf. \textit{M. L. Green}, J. Differ. Geom. 19, 125-171 (1984; Zbl 0559.14008)] based on this technique.
Another case in which the Betti numbers and shifts in the minimal resolution have been studied extensively is the case of points in generic position. The author explains the minimal resolution conjecture (MRC) and the work that has been devoted to it. For \(n+2\leq s\leq n+4\) points in \(\mathbb{P}^n\), the precise geometrical conditions on the points under which MRC holds are described, and sketches of the proofs of those joint results with M. E. Rossi and G. Valla are given.
The paper ends with an explanation of the Green-Lazarsfeld conjecture and an extensive list of references. For anyone interested in syzygies of sets of points in \(\mathbb{P}^n\), this survey can be whole-heartedly recommended as an excellent starting point. coordinate ring of a finite set of points; geometry of the configuration; minimal resolution conjecture Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Syzygies, resolutions, complexes and commutative rings, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series Sets of points and their syzygies | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper under review is the result of an attempt to understand a recent draft of \textit{C.S. Seshadri} [``Moduli spaces of torsion free sheaves and \(G\)-spaces on nodal curves'', preliminary draft, 2002] and is meant as a contribution in the quest for a good compactification of the moduli space (or stack) of \(G-\)bundles on a nodal curve. For finite groups \(G\) the stack of stable maps into \(BG\) has been recently constructed by \textit{D. Abramovich, A. Corti} and \textit{A. Vistoli} [Commun. Algebra 31, No.8, 3547--3618 (2003; Zbl 1077.14034)] by means of the so-called twisted bundles. On the other hand, as shown by the author, the notion of Gieseker vector bundles leads to the construction of the stack of stable maps into \(\text{BGL}_{r}.\)
In this note the author establishes a connection between the straightforward generalization of the notion of twisted bundles to the case of the non-finite reductive group \(\text{GL}_{r}\) and Gieseker vector bundles.
Theorem 3.4. There exists a natural bijection from the set of all Gieseker vector bundle data on \(({\widetilde{C}}, p_{1}, p_{2})\) to the set of all pairs \((E, \Phi),\) where \(E\) is a vector bundle on \(\widetilde{C}\) and \(\Phi\) is a \(k-\)valued point in \(\text{KGL}(E[p_{1}], E[p_{2}]).\) compactification of the moduli space (or stack) of \(G-\)bundles on a nodal curve; twisted bundles; Gieseker vector bundles Vector bundles on curves and their moduli, Algebraic moduli problems, moduli of vector bundles, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Twisted vector bundles on pointed nodal 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 introduce a new class of algebras, called reconstruction algebras, and present some of their basic properties. These non-commutative rings dictate in every way the process of resolving the Cohen-Macaulay singularities \(\mathbb C^2/G\) where \(G=\frac{1}{r}(1,a)\leq\mathrm{GL}(2,\mathbb C)\).
This paper is organized as follows. In Section 2 we define the reconstruction algebra associated to a labelled Dynkin diagram of type \(A\) and describe some of its basic structure. In Section 3 we prove that it is isomorphic to the endomorphism ring of some Cohen-Macaulay modules. In Section 4 the minimal resolution of the singularity \(\mathbb C^2/\frac{1}{r}(1,a)\) is obtained via a certain moduli space of representations of the associated reconstruction algebra \(A_{r,a}\), and in Section 5 we produce a tilting bundle which gives us our derived equivalence. In Section 6 we prove that \(A_{r,a}\) is a prime ring and use this to show that the Azumaya locus of \(A_{r,a}\) coincides with the smooth locus of its centre \(\mathbb C[x,y]^{\frac{1}{r}(1,a)}\). This then gives a precise value for the global dimension of \(A_{r,a}\), which shows that the reconstruction algebra need not be homologically homogeneous. reconstruction algebras; Cohen-Macaulay singularities; labelled Dynkin diagrams; endomorphism rings of Cohen-Macaulay modules; resolutions of singularities; moduli spaces of representations; tilting bundles; derived equivalences; global dimension Wemyss, M, Reconstruction algebras of type \(A\), Trans. Am. Math. Soc., 363, 3101-3132, (2011) Rings arising from noncommutative algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Cohen-Macaulay modules, Representations of quivers and partially ordered sets Reconstruction algebras 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 \(k\) be an algebraically closed field and let \(A\) be a finitely generated \(k\)-algebra (that is, \(A\cong k\langle X_1,X_2,\dots,X_n\rangle/I\), with \(I\) a two-sided ideal). It is known that if \(\varphi\colon A\to B\) is a homomorphism of finitely generated algebras, then there are induced regular \(\text{Gl}_d(k)\)-equivariant morphisms of affine varieties \(\varphi^{(d)}\colon\text{mod}_B(d)\to\text{mod}_A(d)\), for all \(d\geq 1\).
The author deals with the relation between epimorphisms in the category of rings and regular morphisms of module varieties which are immersions. He proves that \(\varphi\colon A\to B\) is an epimorphism in the category of rings if and only if \(\varphi^{(d)}\colon\text{mod}_B(d)\to\text{mod}_A(d)\) is an immersion, for all \(d\geq 1\). He applies this result for classifying the types of singularities (and minimal singularities) in the subvarieties of modules from homogeneous standard tubes in the Auslander Reiten quiver of finite-dimensional algebras. module varieties; regular morphisms; immersions; finitely generated algebras; equivariant morphisms; affine varieties; epimorphisms; minimal singularities; Auslander-Reiten quivers Representations of associative Artinian rings, Group actions on varieties or schemes (quotients), Singularities in algebraic geometry, Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers, Embeddings in algebraic geometry, Automorphisms and endomorphisms Immersions of module varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(M\) be a compact connected Kähler manifold and \(G\) a connected linear algebraic group defined over \({\mathbb{C}}\). A Higgs field on a holomorphic principal \(G\)-bundle \(\mathcal E_{G}\) over \(M\) is a holomorphic section \(\theta\) of \(\text{ad}(\mathcal E_{G})\otimes {\Omega}^{1}_{M}\) such that \(\theta\wedge \theta = 0\). Let \(L(G)\) be the Levi quotient of \(G\) and \((\mathcal E_{G}(L(G)), \theta_{l})\) the Higgs \(L(G)\)-bundle associated with \((\mathcal E_{G}, \theta)\). The Higgs bundle \((\mathcal E_{G}, \theta)\) will be called semistable (respectively, stable) if \((\mathcal E_{G}(L(G)), \theta_{l})\) is semistable (respectively, stable). A semistable Higgs \(G\)-bundle \((\mathcal E_{G}, \theta)\) will be called pseudostable if the adjoint vector bundle \(\text{ad}(\mathcal E_{G}(L(G)\))) admits a filtration by subbundles, compatible with \(\theta\), such that the associated graded object is a polystable Higgs vector bundle. We construct an equivalence of categories between the category of flat \(G\)-bundles over \(M\) and the category of pseudostable Higgs \(G\)-bundles over \(M\) with vanishing characteristic classes of degree one and degree two. This equivalence is actually constructed in the more general equivariant set-up where a finite group acts on the Kähler manifold. As an application, we give various equivalent conditions for a holomorphic \(G\)-bundle over a complex torus to admit a flat holomorphic connection. Pseudostable Higgs bundle; Flat connection; Kähler manifold Biswas, I.; Gómez, T. L., Connections and Higgs fields on a principal bundle, Ann. Glob. Anal. Geom., 33, 19-46, (2008) Vector bundles on curves and their moduli, Group varieties, Holomorphic bundles and generalizations, Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) Connections and Higgs fields on a principal bundle | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(C_0\) be an irreducible, projective algebraic curve over a field of characteristic \(0\). Assume that \(C_0\) has one node and let \(s:C\to C_0\) be a normalization. It is natural to expect that the theory of vector bundles on \(C_0\) is related, via the pull-back, to the theory of vector bundles on \(C\) plus some local data on the two points \(P_1\) and \(P_2\) which map to the node \(P\). The author studies the generalized theta divisors on a compactification of the moduli stack on \(C_0\).
In particular, the author considers the Gieseker compactification, obtained by using vector bundles on some modification of \(C_0\). The corresponding stack GVB of Gieseker vector bundles carries a generalized theta divisor \(\Theta\), which is proven to factorize on some natural extension of the moduli VB of vector bundles on \(C\). This factorization turns out to be canonical.
Namely, if VB is the moduli stack of rank \(n\) vector bundles on \(C\), then there is a natural map \(\text{KGL}\to\text{VB}\), where KGL is an extension of GVB, obtained (roughly speaking) by adding to a vector bundle on \(C_0\) the datum of an isomorphism between the fibers of \(s^*(E)\) at \(P_1,P_2\). The author then considers the variety \(\text{PB}=\text{Fl}(E) \times_{\text{VB}} \text{Fl}(F)\), where Fl is the variety of the full flags in a vector space and \(E,F\) are the fibers of the universal bundle on VB over the sections \(P_1\) and \(P_2\). The relations between KGL an PB yields a canonical decomposition:
\[
H^0(\text{GVB}, \Theta^k) \simeq \oplus_{(a,b)\in A\subset{\mathbb Z}^n\times {\mathbb Z}^n} H^0(\text{PB}, \Theta_{\text{PB}}^k(a,b)).
\]
Kausz, I, A canonical decomposition of generalized theta functions on the moduli stack of gieseker vector bundles, J. Algebraic Geom., 14, 439-480, (2005) Vector bundles on curves and their moduli, Algebraic moduli problems, moduli of vector bundles A canonical decomposition of generalized theta functions on the moduli stack of Gieseker vector bundles | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(F\) be a totally real field of degree \(d\) over \(\mathbb{Q}\), and discriminant \(\Delta\) with ring of integers \(\mathfrak O\) and \(\mathfrak A\) a square-free ideal of \(\mathfrak O\). The group \(SL(2,{\mathfrak O})\) acts on the product of \(d\) copies of the upper half plane \(\mathcal H\) via the \(d\) embeddings of \(SL(2,{\mathfrak O})\) into \(SL(2,\mathbb{R})\) induced by the embeddings of \(F\) into \(\mathbb{R}\). Let \(n\) be an integer relatively prime to \(\mathcal A\) and let \(\Gamma(n)\) denote the kernel of the reduction \(SL(2,{\mathfrak O})\to SL(2,{\mathfrak O}/n{\mathfrak O})\). The modular varieties of the title are \(H_\Gamma={\mathcal H}^d/\Gamma\) where \(\Gamma\) is either
\[
\Gamma_0({\mathfrak A},n)=\Gamma_0({\mathfrak A})\cap \Gamma(n)
\]
or
\[
\Gamma_{00}({\mathfrak A},n)=\Gamma_{00}({\mathfrak A})\cap \Gamma(n)
\]
the subgroups \(\Gamma_0({\mathfrak A}),\Gamma_{00}({\mathfrak A})\) of \(SL(2,{\mathfrak O})\) being defined by
\[
\begin{aligned} & \Gamma_0({\mathcal A})=\left\{{a\;b\choose c\;d}\Bigl|c\in {\mathcal A}\right\}\\ & \Gamma_{00}({\mathcal A})=\left\{{a\;b\choose c\;d}\Bigl|c,a-1, d-1\in {\mathcal A}\right\}.\end{aligned}
\]
The varieties are defined over \(\mathbb{Q}(\zeta_n)\), \(\zeta_n=e^{2\pi i/n}\). The author studies models for \(H_\Gamma\) over \(\text{Spec }\mathbb{Z}[\zeta_n,\frac{1}{n}]\) and in particular the local structure of the reduction at primes, \(p\), \((p,n)=1\), and \(p\mid \text{Norm}({\mathcal A})\). Using the method of integral models due to \textit{M. Rapoport} [Compos. Math. 36, 255-335 (1978; Zbl 0386.14006)], the author describes models for \(H_\Gamma\) that are smooth over \(\mathbb{Z}[\zeta_n,1/(n.\Delta.\text{Norm}({\mathfrak A}))]\), but difficulties arise in the case of models over \(\mathbb{Z}[\zeta_n,\frac1n ]\). In that case when the level of the subgroup is not invertible in the base scheme, it is not clear what the moduli problem should be.
The author sets out to understand the nature of the proper models by considering a notion of \(\mathfrak A\)-level structure for abelian varieties over the primes \(p\), \(p\mid\text{Norm}({\mathfrak A})\). The main result is that the moduli spaces are normal and relative complete intersections and the singularities are the same for Shimura varieties associated to forms of the reductive group. -- The author also obtains a construction of regular scheme models for certain Hilbert-Blumenthal surfaces of the form \(H_{\Gamma_{00}({\mathcal A},1)}\) over \(\text{Spec }\mathbb{Z}\) with no prime inverted. Some of the problems that arise in the cases of non-square-free level are mentioned, but not dealt with.
The paper uses the language of algebraic stacks [see \textit{P. Deligne} and \textit{D. Mumford}, Publ. Math., Inst. Hautes Étud. Sci. 36, 75-109 (1969; Zbl 0181.48803)], but the reader who is unfamiliar with the language (as was the reviewer until he read this paper and that one -- whence the unconscionable delay in the review itself) can assume that \(n\geq 3\), in which case the results refer to algebraic spaces or schemes, but the effort to understand the background is well worth while. Hilbert modular varieties; Shimura varieties; Hilbert-Blumenthal surfaces; algebraic stacks G. Pappas, Arithmetic models for Hilbert modular varieties. Compositio Math. 98 (1995), 43-76. Modular and Shimura varieties, Algebraic moduli of abelian varieties, classification Arithmetic models for Hilbert modular 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 Our investigation in the present paper is based on three important results. (1) In [14], Ringel introduced Hall algebra for representations of a quiver over finite fields and proved the elements corresponding to simple representations satisfy the quantum Serre relation. This gives a realization of the nilpotent part of quantum group if the quiver is of finite type. (2) In [6], Green found a homological formula for the representation category of the quiver and equipped Ringel's Hall algebra with a comultiplication. The generic form of the composition subalgebra of Hall algebra generated by simple representations realizes the nilpotent part of quantum group of any type. (3) In [11], Lusztig defined induction and restriction functors for the perverse sheaves on the variety of representations of the quiver which occur in the direct images of constant sheaves on flag varieties, and he found a formula between his induction and restriction functors which gives the comultiplication as algebra homomorphism for quantum group. In the present paper, we prove the formula holds for all semisimple complexes with Weil structure. This establishes the categorification of Green's formula. quiver; perverse sheaf; restriction functor; Green's formula The parity of Lusztig's restriction functor and Green's formula | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this paper, the authors develop the concept of \(\mathbf{q}\)-CW complex structure on an orbifold, in which the analog of an open cell is the quotient of an open disk by an action of a finite group (i.e the so called \(\mathbf{q}\)-cell), to detect torsion in its integral cohomology. As the ordinary CW complex, the \(\mathbf{q}\)-CW complex is constructed inductively, from a discrete set, by attaching finitely many \(\mathbf{q}\)-cells to lower dimensional skeleton. Let \(X\) be a finite \(\mathbf{q}\)-CW complex with no odd dimensional \(\mathbf{q}\)-cells.
The main result about \(X\) is: If a prime \(p\) is co-prime to the orders of all the finite groups \(G_i\), which appear in the \(\mathbf{q}\)-cells of \(X\), then \(H_*(X;\mathbb Z)\) has no \(p\)-torsion and \(H_{\mathrm{odd}}(X;\mathbb Z_p)\) is trivial.
Successive applications of the above result yield another impotent Theorem 1.2. Then, the authors use the main results to improve upon certain previous results about toric orbifolds, toric varieties and weighted Grassmannians. In the last section, the the general \(\mathbf{q}\)-CW complexes which do not necessarily consists of even dimensional \(\mathbf{q}\)-cells is discussed. Similar results are obtained, under the hypothesis that the prime \(p\) is coprime to the nonzero degrees of the attaching maps of the \(\mathbf{q}\)-CW complex. Generalizations (algebraic spaces, stacks), Toric varieties, Newton polyhedra, Okounkov bodies, Complex spaces with a group of automorphisms, Orbifold cohomology On integral cohomology of certain orbifolds | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(Q\) be a tame connected quiver (i.e. the underlying graph \(|Q|\) of \(Q\) is a Dynkin or extended Dynkin diagram), and let \(d\) be a prehomogeneous dimension vector (i.e. the space of representations of \(Q\) with dimension vector \(d\) contains a Zariski open \(\text{GL}(d)\)-orbit). Denote by \(Z_{Q,d}\) the closed subscheme in the representation space of the common zeros of the basic semi-invariant polynomial functions \(f_1,\dots,f_s\) (the null-cone); recall that the zero sets \(Z(f_1),\dots,Z(f_s)\) are the codimension \(1\) irreducible components of the complement of the open orbit.
In an earlier paper, \textit{Ch. Riedtmann} and \textit{G. Zwara} [Comment. Math. Helv. 79, No. 2, 350-361 (2004; Zbl 1063.14052)] showed that \(Z_{Q,d}\) is a complete intersection if \(|Q|=A_n\) or \(\widetilde A_n\).
The main result of the present paper is that \(Z_{Q,d}\) is not too far from being a complete intersection for all tame prehomogeneous quiver settings. More precisely, it is proved that \(s-\text{codim}(Z_{Q,d})\leq\gamma(|Q|)\), where \(\gamma(|Q|)\in\{0,1,2,3,4\}\) is explicitly given for each (extended) Dynkin diagram, and the bound is sharp with the possible exception of the case of \(\widetilde E_8\). The proof uses the Auslander-Reiten theory for representations of tame quivers. (Also submitted to MR.) tame quivers; prehomogeneous dimension vectors; open orbits; common zero loci of semi-invariants; extended Dynkin diagrams; semi-invariant polynomials Riedtmann, Ch, Tame quivers, semi-invariants, and complete intersections, J. Algebra, 279, 362-382, (2004) Representations of quivers and partially ordered sets, Complete intersections, Geometric invariant theory, Vector and tensor algebra, theory of invariants Tame quivers, semi-invariants, and complete intersections. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 the abelian-nonabelian correspondence for quasimap \(I\)-functions. That is, if \(Z\) is an affine l.c.i. variety with an action by a complex reductive group \(G\), we prove an explicit formula relating the quasimap \(I\)-functions of the geometric invariant theory quotients \(Z\mathord{/\mkern -6mu/}_{\theta}G\) and \(Z\mathord{/\mkern -6mu/}_{\theta}T\) where \(T\) is a maximal torus of \(G\). We apply the formula to compute the \(J\)-functions of some Grassmannian bundles on Grassmannian varieties and Calabi-Yau hypersurfaces in them. Geometric invariant theory, Grassmannians, Schubert varieties, flag manifolds, Stacks and moduli problems, Formal methods and deformations in algebraic geometry, Sheaves in algebraic geometry The abelian-nonabelian correspondence for \(I\)-functions | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(k\) be an algebraically closed field. For a quiver \(Q=(Q_0 ,Q_1, s, e)\) and a dimension vector \(d\in\mathbb{Z}^{Q_0}\) we let rep\(_Q(d)\) be the vector space of representations corresponding to \(d\). Now \(\text{GL}(d)\) acts on rep\(_Q(d)\) in a natural way, hence given a representation \(V\) or \(Q\) we can consider the \(\text{GL}(d)\)-orbit in rep\(_Q(d),\) which consists of the representations isomorphic to \(V\) -- we shall write this orbit as \(\mathcal{O}_V.\) Let \(M\) and \(N\) be representations in rep\(_Q(d)\) such that \(\mathcal{O}_N\subset\mathcal{\bar{O}}_M,\) where \(\mathcal{\bar{O}}_M\) is the Zariski closure of \(\mathcal{O}_M\) in rep\(_Q(d).\) Write Sing\((M,N)\) for the set of smoothly equivalent classes of pointed varieties. The regular points form a type of singularity which we shall denote Reg. In the case where \(\mathcal{O}_N\) has codimension one in \(\mathcal{\bar{O}}_M\) it is known that Sing\((M,N) =\) Reg; furthermore of \(\mathcal{O}_N\) has codimension two and \(Q\) is a Dynkin quiver then again Sing\((M,N) =\) Reg.
Here the author investigates the codimension two case when \(Q\) is an extended Dynkin quiver. In the case where \(Q\) is the Kronecker quiver one has two types of singularities, namely \(A_r=\) Sing\((\mathcal{A}_{r+1},0)\) and \(C_r=\) Sing\((\mathcal{C}_r,0)\), where \(\mathcal{A}_r=\{ ( uv, u^r, v^r) \in k^3\mid u,v\in k\} \) and \(\mathcal{C}_r=\{(u^r, u^{r-1}v,\dots,v^r) \in k^{r+1}\mid u,v\in k\} .\) Note that \(C_1=\) Reg, \(C_2=A_1\) and the remaining types are all distinct. The main result is that for \(Q\) an extended Dynkin quiver in the codimension two case we have Sing\((M,N) \) is one of the \(A_r\) or \(C_r\)'s. If \(Q\) is a cyclic quiver then Sing\((M,N) \neq C_r\) for \(r\geq3,\) i.e. Sing\((M,N) =A_r\) or Reg. Dynkin quivers; representations of quivers; singularities of representations of quivers Zwara, G.: Codimension two singularities for representations of extended Dynkin quivers. Manuscr. Math. 123(3), 237--249 (2007) Singularities in algebraic geometry, Group actions on varieties or schemes (quotients), Representations of quivers and partially ordered sets Codimension two singularities for representations of extended 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 \(K\) be an algebraically closed field and let \(Q=(Q_0, Q_1)\) be a quiver (i.e. a finite oriented graph), where \(Q_0\) is the set of vertices of \(Q\) and \(Q_1\) is the set of arrows of \(Q\). We denote by \(KQ\) the path \(K\)-algebra of \(Q\). If \(X\) is a finitely generated right \(KQ\)-module and \(\beta\in\mathbb{N}^{Q_0}\) is a dimension vector we denote by \(\text{Gr}_{KQ}(X,\beta)\) the algebraic Grassmann variety of all submodules \(Y\) of \(X\) of codimension \(\beta\), that is, the dimension vector of \(X/Y\) is \(\beta\).
Let \(b_Q(-,-):\mathbb{Z}^{Q_0}\times\mathbb{Z}^{Q_0}\to\mathbb{Z}\) be the bilinear form associated with \(Q\). Given a dimension vector \(\beta\in\mathbb{N}^{Q_0}\) we call the affine variety
\[
\text{Rep}_{KQ}(\beta)=\prod_{\gamma:i\to j}\text{Hom}_K(K^{\beta(i)},K^{\beta(j)})
\]
the configuration space of representations of \(Q\) of dimension vector \(\beta\). Given \(y\in\text{Rep}_{KQ}(\beta)\) we denote by \(K_y\) the right \(KQ\)-module corresponding to \(y\). By a rank of a \(KQ\)-homomorphism \(X\to K_y\) we mean the dimension vector of its image. If \(X\) is a finitely generated right \(KQ\)-module, \(\beta\in\mathbb{N}^{Q_0}\) is a dimension vector and \(y\in\text{Rep}_{KQ}(\beta)\) then \(\text{Hom}_{KQ}(X,K_y)\) is a finite dimensional vector space and the function \(y\mapsto\dim\text{Hom}_{KQ}(X,K_y)\) is an upper semicontinuous function on \(\text{Rep}_{KQ}(\beta)\). It follows that the minimum value of this function, denoted by \(\text{hom}(X,\beta)\), is also its general value. For any \(\alpha\in\mathbb{N}^{Q_0}\) the set of all homomorphisms \(X\to K_y\) of rank at most \(\alpha\) is a closed subset of \(\text{Hom}_{KQ}(X,K_y)\). It follows that there is a unique maximal rank \(\gamma_{X,y}\) of homomorphisms \(X\to K_y\), and that the set of homomorphisms of rank \(\gamma_{X,y}\) is an open subset of \(\text{Hom}_{KQ}(X,K_y)\). The function \(y\mapsto\gamma_{X,y}\) is constant on a non-empty open subset of \(\text{Rep}_{KQ}(\beta)\), and its general value is denoted by \(\gamma_{X,\beta}\).
One of the main results of the paper asserts that if \(X\) is a finite dimensional right \(KQ\)-module and \(\beta\in\mathbb{N}^{Q_0}\) is a dimension vector then \(\text{hom}(X,\beta)=b_Q(\gamma_{X,\beta},\beta)-\dim U\) for some open subset \(U\) of the Grassmann variety \(\text{Gr}_{KQ}(X,\gamma_{X,\beta})\).
As a consequence the author derives the following two corollaries concerning the asymptotic behaviour of \(\text{hom}(X,\beta)\) as \(\beta\) increases: (a) If \(X\) is a finitely presented right \(KQ\)-module and \(\beta\in\mathbb{N}^{Q_0}\) is a dimension vector then
\[
\lim_{r\to\infty}\text{hom}(X,r\beta)=\max\{\widetilde\beta(X/Y)\mid Y\subseteq X\text{ a finitely presented submodule\}}
\]
where \(\widetilde\beta:K_0(KQ)\to\mathbb{Z}\) is the natural homomorphism induced by \(\beta\). (b) If \(\beta\in\mathbb{N}^{Q_0}\) is a dimension vector and \(X\) is a finitely presented right \(KQ\)-module which is \(\beta\)-semistable then \(\text{hom}(X,r\beta)=0\). These results are related to a work of A. Schofield on homological epimorphisms from the path algebra \(KQ\) to a simple artinian ring. path algebras; maximal rank of homomorphisms; quivers; finitely generated right modules; dimension vectors; Grassmann varieties; bilinear forms; affine varieties; configuration spaces of representations; finitely presented right modules; simple Artinian rings Crawley-Boevey, W, On homomorphisms from a fixed representation to a general representation of a quiver, Trans. Am. Math. Soc., 348, 1909-1919, (1996) Representations of quivers and partially ordered sets, Grassmannians, Schubert varieties, flag manifolds, Algebraic theory of abelian varieties, Vector and tensor algebra, theory of invariants On homomorphisms from a fixed representation to a general representation of a quiver | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0511.00010.]
In these notes I will discuss two approaches to the study of the orbits, invariants, etc. of a linear reductive group G operating on a finite dimensional vector space V. The two techniques are the ''quiver method'' and the ''slice method'', which are discussed in chapters I and II respectively. -- Undoubtedly, the slice method, based on Luna's slice theorem [\textit{D. Luna,} Bull. Soc. Math. Fr., Suppl., Mém. 33, 81-105 (1973; Zbl 0286.14014)], is one of the most powerful methods in geometric invariant theory. Even in the case of binary forms the slice method gives results which were out of reach of mathematics of 19th century. For example, I show that for the action of \(SL_ 2({\mathbb{C}})\) on the space of binary form of odd degree \(d>3\) the minimal number of generators of the algebra of invariant polynomials is greater than p(d-2), where p(n) is the classical partition function. On the other hand, the quiver method can be applied to a (very special) class of representations for which the slice method often fails. Most of the results of chapter I are contained in the author's paper in J. Algebra 78, 141-162 (1982; Zbl 0497.17007) and in C. R. Acad. Sci., Paris, Sér. A 283, 875-878 (1976; Zbl 0343.20023) by the author, \textit{V. L. Popov} and \textit{E. B. Vinberg}; on the most part I just give simpler versions of the proofs. Chapter II contains some new results (as, it seems, the one mentioned above). linear reductive group operating on a vector space; invariants; quiver method; slice method V. G. Kac, ''Root systems, representations of quivers and invariant theory,'' in Invariant Theory, New York: Springer-Verlag, 1983, vol. 996, pp. 74-108. Geometric invariant theory, Group actions on varieties or schemes (quotients) Root systems, representations of quivers and invariant theory | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The problem treated in this paper was posed by J. Nash, who proposed to study the space \({\mathcal H}\) of arcs in a germ of a singular variety \((V,P)\); he showed that the arc families in \({\mathcal H}\) correspond to irreducible components of the exceptional locus of any given desingularization of \((V,P)\) and raised the question whether every component which (modulo birational equivalence) appears in every desingularization (such components are called essential) is associated to some family in \({\mathcal H}\).
The paper gives an affirmative answer in the case when \(V\) is a surface and the singularity is rational. In this case the essential components \(\{E_\alpha\}_{\alpha\in \Delta}\) are represented by the irreducible components of a minimal desingularization \(S\to V\). -- The idea is to define, for each \(E_\alpha\), a \(\mathbb{Q}\)-Cartier divisor \(D_\alpha\) in \(S\) (\(D_\alpha\) is the exceptional part of the total transform of the projection of any germ of curve in \(S\) which is transversal to \(E_\alpha\) and does not meet any \(E_\gamma\), \(\gamma\neq \alpha\)); then a closed subset \({\mathcal H}_\alpha\) of \({\mathcal H}\) is defined by imposing valuative conditions determined by \(D_\alpha\), and those \({\mathcal H}_\alpha\) give the answer to the Nash problem. rational singularity; singular variety; desingularization A. -J. Reguera, ''Families of arcs on rational surface singularities,'' Manuscripta Math., vol. 88, iss. 3, pp. 321-333, 1995. Singularities of surfaces or higher-dimensional varieties, Global theory and resolution of singularities (algebro-geometric aspects) Families of arcs on rational surface singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let G be a simply connected simple algebraic group defined over an algebraic number field K, and let S be a finite subset of the set \(V^ K\) of valuations of K which contains the set \(V^ K_{\infty}\) of Archimedean valuations. Let \(G(K)^{\wedge}\) (\(\overline{G(K)}\), resp.) denote the completion of the group of K-rational points G(K) with respect to the S-arithmetic topology (the S-congruence topology, resp.). The kernel of the natural projection \(\pi\) : G(K)\({}^{\wedge}\to \overline{G(K)}\) is denoted by C(S,G), and called the congruence kernel. \textit{J.-P. Serre} [Ann. Math., II. Ser. 92, 489-527 (1970; Zbl 0239.20063)] made the following ``congruence conjecture'': If \(rank_ SG=\sum_{v\in S}rank_{K_ v}G\geq 2\) and if G is isotropic over \(K_ v\) for all \(v\in S\setminus V^ K_{\infty}\), then the congruence kernel C(S,G) is finite. By developing the methods of \textit{M. Kneser} [J. Reine Angew. Math. 311/312, 191-214 (1979; Zbl 0409.20038)] and bringing in some new ideas, the author obtains a proof of the congruence conjecture for all groups of classical types which have a geometrical realization of sufficiently large degree. The geometrical method used there turns out not to be applicable to most of the exceptional groups. By using the inner structure of the group, the author shows that, if G is an anisotropic K-group of one of the types \(E_ 7\), \(E_ 8\), \(F_ 4\), or \(G_ 2\), then the congruence kernel is trivial for \(rank_ SG\geq 2\). Groups of type \(E_ 6\) are not included here. simply connected simple algebraic group; algebraic number field; valuations; completion; group of K-rational points; S-arithmetic topology; S-congruence topology; congruence kernel; congruence conjecture; exceptional groups; anisotropic K-group Rapinchuk A S, On the congruence subgroup problem for algebraic groups,Dokl. Akad. Nauk. SSSR 306 (1989) 1304--1307 Linear algebraic groups over global fields and their integers, Unimodular groups, congruence subgroups (group-theoretic aspects), Classical groups (algebro-geometric aspects) On the congruence problem for algebraic groups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G \subset \text{GL}(2,\mathbb{C})\) be a finite subgroup, and \(Y=G\text{-Hilb}(\mathbb{C}^2)\) be the Hilbert scheme of \(G\)-clusters \(\mathcal{Z}\subset \mathbb{C}^2 \). By definition \(\mathcal{Z}\) is a 0-dimensional subscheme of \(\mathbb{C}^2\) of length \(|G|\) such that \(H^0(\mathcal{O}_{\mathcal{Z}})\) is isomorphic to the regular representation of \(G\). It is known that \(Y\) is isomorphic to the minimal resolution of the singularity \( \mathbb{C}^2/G \).
The paper under review gives an explicit description of a natural affine open covering \(Y\) in the case that \(G\) is small binary dihedral group. The open covering is in bijection with the set of bases for \(H^0(\mathcal{O}_{\mathcal{Z}})\) which are called the \(G\)-graphs. All the possible \(G\)-graphs are explicitly calculated by interpreting the action of \(G\) as the cyclic action of its maximal normal abelian subgroup \(H\) of index 2 followed by a dihedral involution. As an application the classification of the \(G\)-graphs is used to list the special representations of any small binary dihedral group. \(G\)-graphs; special representations; binary dihedral groups; McKay correspondence Alvaro Nolla de Celis, \(G\)-graphs and special representations for binary dihedral groups in \(\mathrm{GL}(2,\mathbf{C})\). (to appear in Glasgow Mathematical Journal). McKay correspondence, Parametrization (Chow and Hilbert schemes) \(G\)-graphs and special representations for binary dihedral groups in \(\mathrm{GL}(2,\mathbb C)\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \((X,G)\) denote a regular action of an affine algebraic group on an affine algebraic variety \(X\) over an algebraically closed field \(K\), \(\mathrm{char}(K)=p \geq 0\). Such an action \((X,G)\) is said to be \textit{equidimensional} ( resp. \textit{cofree}), if the quotient morphism \(\pi_{X,G}:X \to X/\!/G\) is equidimensional, i.e., closed fibers of \(\pi_{X,G}\) are of dimension \(\dim X- \dim X/\!/G\) (resp. if \(\mathcal{O}(X)\) is \(\mathcal{O}(X)^G\)-free). The action \((X,G)\) are called \textit{conical} if the action of \(G\) preserves each homogeneous part of \(\mathcal{O}(X).\) Moreover, \((X,G)\) is said to be \textit{stable}, if there is a non-empty open subset of \(X\) consists of closed \(G\)-orbits. Let \(\mathrm{Cl}(X) \) denotes the divisor class group of \(X\). For a finite dimensional complex linear representation \(G \to \mathrm{GL}(V)\) of a connected algebraic group \(G\) the following conjecture is well known
Russian conjecture. \textit{If \((V,G)\) is equidimensional, then it is cofree.}
Especially for an algebraic torus, the Russian conjecture is solved affirmatively [\textit{D. Wehlau}, A proof of the Popov conjecture for tori. Proc. Am. Math. Soc. 114, No.~3, 839--845 (1992; Zbl 0754.20013)], which is generalised by the author to the case where \((V,G)\) is a conical factorial variety \(V=X\) with a stable conical action of an algebraic torus over \(K\) of characteristic zero. The author study the following problem which produces extensions of the results in [\textit{H. Nakajima}, ``Equidimensional actions of algebraic tori'', Ann. Inst. Fourier 45, No.~3, 681--705 (1995; Zbl 0823.14035)].
Problem. Suppose that \(G\) is an algebraic torus and \((X,G)\) is a stable conical action of \(G\) on a conical normal variety \(X\) defined over \(K\) of characteristic zero. If \((X,G)\) is equidimensional, then
-- Does there exist a \(G\)-equivariant finite Galois covering \(X \to \tilde{X}\) for a normal conical variety \(\tilde{X}\) with a conical \(G\)-action admitting some commutative diagram such that \((\tilde{X},G)\) is cofree?
-- Moreover can we choose \(X \to \tilde{X}\) in such a way that the order \(|\mathrm{Gal}(X/\tilde{X})|\) of the group of \(X \to \tilde{X}\) is a divisor of a power of the exponent of a subgroup pf the divisor group \(\mathrm{Cl}(X) \)?
Let \((X,T)\) be a regular stable conical action of an algebraic torus on an affine normal conical variety \(X\). The author defined a certain subgroup of \(\mathrm{Cl}(X /\!/T)\) and characterize its finiteness in terms of a finite \(T\)-equivariant Galois descent \(\tilde{X}\) of \(X\). The author shows that the action \((X,T)\) is equidimensional if and only if there exists a \(T\)-equivariant finite Galois covering \(X \to \tilde{X}\) such that \((\tilde{X},T)\) is cofree. Moreover the order of \(\mathrm{Cl}(X /\!/\tilde{X})\) is controlled by a certain subgroup of \(\mathrm{Cl}(X ).\) The present result extends thoroughly the equivalence of equidimensionality and cofreenes of \((X,T)\) for a factorial \(X.\) The purpose of the paper is to evaluate orders of divisor classes associated to modules of relative invariants for a Krull domain with a group action. This is useful in studying on equidimensional torus action as above. relative invariant; equidimensional action; cofree action; divisor class group; algebraic torus Nakajima, H., Reduced class groups grafting relative invariants, Adv. Math., 227, 920-944, (2011) Actions of groups on commutative rings; invariant theory, Group actions on affine varieties, Group actions on varieties or schemes (quotients), Toric varieties, Newton polyhedra, Okounkov bodies, Representation theory for linear algebraic groups, Linear algebraic groups over arbitrary fields Reduced class groups grafting relative 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 of \textit{S. M. Gusein-Zade, F. Delgado} and \textit{A. Campillo} [Funct. Anal. Appl. 33, No. 1, 56--57 (1999); translation from Funkts. Anal. Prilozh. 33, No. 1, 66--68 (1999; Zbl 0967.14017] initiated an intense activity regarding different multi-variable filtrations associated with divisorial filtrations of exceptional divisors of different resolutions.
In the present article, the author defines a Poincaré series associated with a germ of a toric or analytically irreducible quasi-ordinary hypersurface singularity via a finite sequence of monomial valuations, provided that at least one of them is centered at the origin. This involves the definition of a multi-graded ring associated with the analytic algebra of the singularity by the sequence of the valuations. The author proves that the corresponding Poincaré series is a rational function with integer coefficients. Moreover, it can also recovered as an integral with respect to the Euler characteristic of a function defined (by the valuations) on the projectivization of the analytic algebra of the germ. This shows that the Poincaré series associated with the set of divisorial valuations of the essential divisors is an analytic invariant of the singular germ.
In the quasi-ordinary hypersurface case, the author shows that the Poincaré series is equivalent with the normalized sequence of characteristic monomials (in the analytic case, this set is a complete invariant of the embedded topological type). quasi-ordinary singularities; Poincaré series; multi-graded rings; valuations; divisorial valuations; characteristic monomials; hypersurface singularities; Nash map; toric singularities Singularities of surfaces or higher-dimensional varieties, Invariants of analytic local rings, Toric varieties, Newton polyhedra, Okounkov bodies Quasi-ordinary singularities, essential divisors and Poincaré series | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities A classical result of Gabriel implies that two algebraic varieties are isomorphic if they have equivalent categories of coherent sheaves. How much can be said if the varieties are no longer algebraic?
The paper under review gives a precise answer to this question for generic \(K3\) surfaces and generic compact complex tori. The main result of this article says, in stark contrast to Gabriel's result, that for such manifolds, the category of coherent sheaves does not depend on the complex structure. A complex structure on a \(K3\) surface or on a compact torus of dimension \(n\geq 2\) is said to be generic if it does not have non-trivial integral \((p,p)\)-classes for \(0<p<2n\). In particular, such manifolds are not algebraic.
For the proofs, the author makes essential use of the fact that the manifolds considered are holomorphically symplectic, hence carry a hyperkähler structure. The corresponding twistor space comes as a \(\mathbb{P}^1\)-family of complex structures on the underlying manifold. The generic complex structures form a dense subset of this \(\mathbb{P}^1\) and have a countable complement. A result of the author [Geom. Funct. Anal. 6, No. 4, 601--611 (1996; Zbl 0861.53069)] says that any two points in the same connected component of the moduli space of complex structures on a compact hyperkähler manifold can be connected by a chain of hyperkähler \(\mathbb{P}^1\)'s, such that their intersection points represent generic complex structures. This reduces the proof of the main result to showing the required equivalence for any two generic complex structures in the same hyperkähler family.
The proof of the main result uses the twistor correspondence which associates a holomorphic vector bundle on the twistor space to a vector bundle with a connection with \(\text{SU}(2)\)-invariant curvature on the hyperkähler manifold. In a first step, he proves that a certain subcategory of the category of reflexive sheaves on the twistor space is equivalent to the category of reflexive sheaves (i.e.\ bundles) on the given manifold, equipped with any generic complex structure from the hyperkähler family. In the final step of the proof, singularities are incorporated into the above picture: a certain subcategory of the category of coherent sheaves on the twistor space is shown to be equivalent (via restriction) to the category of coherent sheaves for any generic complex structure in the hyperkähler family. An important ingredient into the proofs is the result that on generic \(K3\) surfaces and tori, reflexive sheaves are automatically locally free and coherent sheaves have isolated singularities only. category of coherent sheaves; reflexive sheaf; hyperkähler manifold; holomorphic symplectic manifold; Hermite-Einstein bundle; twistor space; twistor correspondence; reflexive sheaves on Cohomology of compact hyperkähler manifolds and its applications Verbitsky, M., Coherent sheaves on general \textit{K}3 surfaces and tori, Pure Appl. Math. Q., 4, 3, part 2, 651-714, (2008) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Vector bundles on surfaces and higher-dimensional varieties, and their moduli, \(K3\) surfaces and Enriques surfaces, Calabi-Yau manifolds (algebro-geometric aspects), Twistor theory, double fibrations (complex-analytic aspects), Analytic sheaves and cohomology groups Coherent sheaves on general \(K3\) surfaces and tori | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let $X$ be a fine and saturated log scheme over a field $k$. The paper under review studies two main objects associated to $X$: the infinite root stack $\sqrt[\infty]X$ and the valuativization $X^{\text{val}}$. The infinite root stack $\sqrt[\infty]X$ is obtained by taking inverse limit of $\sqrt[n]X$, the algebraic stack obtained by extracting all the $n$th roots of the divisors in the non-trivial locus of the log structure of $X$. $X^{\text{val}}$ is obtained by taking inverse limit of all log blow-ups $X_\mathcal{I}$ of $X$. Both of these are pro-objects in stacks and their categories of perfect complexes are defined as colimits of dg-categories of perfect complexes over $\sqrt[n]X$ and $X_\mathcal{I}$. \par Now suppose that $X$ is equipped with ``a simple log semistable morphism'' of log schemes $f : X \to S$ for a log flat base $S$. This means that $f$ is log smooth and vertical, and for every geometric point $x$ of $X$ with image $s$ and with non-trivial log structure, there are isomorphisms $\overline M_{S,s} \cong \mathbb N$ and $\overline M_{X,x} \cong \mathbb{N}^{r+1}$ with $r \ge 0$, such that the map $\overline M_{S,s}\to \overline M_{X,x}$ is identified with the $r$-fold diagonal map $\mathbb N \to \mathbb N^{r+1}$. \par The first main result of the paper (viewed as a global McKay correspondence) proves an equivalence of dg-categories of perfect complexes $\text{Perf}(X^{\text{val}}_\infty)\cong \text{Perf}(\sqrt[\infty]X)$, where $X_\infty$ is pullback of the family $f: X\to S$ via the natural map $\sqrt[\infty]S \to S$. In the case that $S$ is the spectrum of a DVR, and $0 \in \sqrt[\infty]S$ is the closed point the equivalence above restricts to an equivalence $\text{Perf}((X^{\text{val}}_\infty)_0)\cong \text{Perf}((\sqrt[\infty]X)_0)$ over the central fibers. \par The next main result of the paper can be viewed as a categorified excision for parabolic sheaves: let $(Y, D)$ be a pair given by a smooth variety $Y$ over $k$ equipped with a normal crossings divisor $D$. Let $(Y',D') \to (Y, D)$ be a log blow-up, such that $(Y',D')$ is again a smooth variety with a normal crossings divisor. Then there is an equivalence $D^b(\text{Par}(Y, D)) \cong D^b(\text{Par}(Y',D'))$ between the bounded derived categories of coherent parabolic sheaves with rational weights. McKay correspondence; semistable degenerations; parabolic sheaves Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Fibrations, degenerations in algebraic geometry, McKay correspondence On a logarithmic version of the derived McKay correspondence | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We show that there exists a morphism between a group \(\Gamma ^{alg}\) introduced by \textit{G. Wilson} [Invent. Math. 133, No. 1, 1--41 (1998; Zbl 0906.35089)] and a quotient of the group of tame symplectic automorphisms of the path algebra of a quiver introduced by \textit{R. Bielawski} and \textit{V. Pidstrygach} [Adv. Math. 226, No. 3, 2796--2824 (2011; Zbl 1220.14011)]. The latter is known to act transitively on the phase space \(C_{n,2}\) of the Gibbons-Hermsen integrable system of rank 2, and we prove that the subgroup generated by the image of \(\Gamma ^{alg}\) together with a particular tame symplectic automorphism has the property that, for every pair of points of the regular and semisimple locus of \(C_{n,2}\), the subgroup contains an element sending the first point to the second. Gibbons-Hermsen system; quiver variety; noncommutative symplectic geometry; integrable system Mencattini, Igor; Tacchella, Alberto: A note on the automorphism group of the bielawski-pidstrygach quiver, Sigma 9, No. 037 (2013) Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Representations of quivers and partially ordered sets, Symplectic manifolds (general theory), Noncommutative geometry (à la Connes), Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) A note on the automorphism group of the Bielawski-Pidstrygach quiver | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Symmetric obstruction theory arises as a curve counting on a Nakajima's quiver variety (\S3.3. cf [\textit{K. Behrend} and \textit{B. Fantechi}, Algebra Number Theory 2, No. 3, 313--345 (2008; Zbl 1170.14004)]). Two classes of moduli examples which carry symmetric obstruction theories already known ([\textit{R. Pandharipande} and \textit{R. P. Thomas}, Invent. Math. 178, No. 2, 407--447 (2009; Zbl 1204.14026)], [\textit{R. P. Thomas}, J. Differ. Geom. 54, No. 2, 367--438 (2000; Zbl 1034.14015)], [\textit{B. Szendrői}, Geom. Topol. 12, No. 2, 1171--1202 (2008; Zbl 1143.14034)]).
In this paper, taking \(\mathbf{M}\), a certain collection of pairs of line bundles on a projective smooth curve \(C\) such that the product of each pair is isomorphic to the canonical sheaf \(\omega_C\), moduli of coherent \(\mathbf{M}\)-twisted quiver sheaves on \(C\) associated to double quivers with moment map relations is shown a new example that carry symmetric obstruction theories.
To define \(\mathbf{M}\) and the moduli \(QG_\beta\), \(\beta\) is a fixed degree class and \(QG_{\mathbf{M},\beta}\), symplectic quotient and stable quasi map are explained in \S2 and 3. \(QG_\beta\) is a finite type DM stack by the bounded theorem ([\textit{I. Ciocan-Fontanine} et al., J. Geom. Phys. 75, 17--47 (2014; Zbl 1282.14022)]). If \(C\) is an elliptic curve, \(QG_\beta\) is shown to carry symmetric obstruction theories by this fact (Theorem 3.5). For general \(C\), discussions on twisted quasimaps are needed (\S4.2). Then the moduli in the case general \(C\) is shown to carry symmetric obstruction theories (Theorem 4.3).
The typical examples to which Theorem 4.3 is applied are Nakajiam's quiver varieties. This is explained in \S5. The stability used in this paper can be replaced as a Rudakov's stability condition on a suitable abelian category of representations of the path algebra \(\mathbb{C}\bar{Q}\) with relations ([\textit{A. Rudakov}, J. Algebra 197, No. 1, 231--245 (1997; Zbl 0893.18007)]). This is explained in \S6, the last Section. Using this replacement, , the moduli space \(QC_{\delta,\mathbf{M},\beta}\) of \(\mathbf{M}\)-twisted quiver bundles are shown to be an algebraic space of finite type over \(\mathbb{C}\) (Theorem 6.6). GIT; stable quasimaps; holomorphic symplectic quotients; twisted quiver bundles; symmetric obstruction theory B. Kim, Stable quasimaps to holomorphic symplectic quotients. arXiv:1005.4125. Vector bundles on curves and their moduli, Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Algebraic moduli problems, moduli of vector bundles, Representations of quivers and partially ordered sets, Complex-analytic moduli problems Stable quasimaps to holomorphic symplectic quotients | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper is the first in a series in which we offer a new framework for hermitian K-theory in the realm of stable \(\infty\)-categories. Our perspective yields solutions to a variety of classical problems involving Grothendieck-Witt groups of rings and clarifies the behaviour of these invariants when 2 is not invertible. In the present article we lay the foundations of our approach by considering Lurie's notion of a Poincaré \(\infty\)-category, which permits an abstract counterpart of unimodular forms called Poincaré objects. We analyse the special cases of hyperbolic and metabolic Poincaré objects, and establish a version of Ranicki's algebraic Thom construction. For derived \(\infty\)-categories of rings, we classify all Poincaré structures and study in detail the process of deriving them from classical input, thereby locating the usual setting of forms over rings within our framework. We also develop the example of visible Poincaré structures on \(\infty\)-categories of parametrised spectra, recovering the visible signature of a Poincaré duality space. We conduct a thorough investigation of the global structural properties of Poincaré \(\infty\)-categories, showing in particular that they form a bicomplete, closed symmetric monoidal \(\infty\)-category. We also study the process of tensoring and cotensoring a Poincaré \(\infty\)-category over a finite simplicial complex, a construction featuring prominently in the definition of the L- and Grothendieck-Witt spectra that we consider in the next instalment. Finally, we define already here the 0th Grothendieck-Witt group of a Poincaré \(\infty\)-category using generators and relations. We extract its basic properties, relating it in particular to the 0th L- and algebraic K-groups, a relation upgraded in the second instalment to a fibre sequence of spectra which plays a key role in our applications. Grothendieck-Witt groups; Hermitian \(K\)-theory; \(L\)-theory; Poincaré categories Hermitian K-theory for stable \(\infty\)-categories. I: Foundations | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We study Schlichting's \(K\)-theory groups of the Buchweitz-Orlov singularity category \(\mathcal D^{\mathrm{sg}}(X)\) of a quasiprojective algebraic scheme \(X/k\) with applications to algebraic \(K\)-theory. We prove for isolated quotient singularities over an algebraically closed field of characteristic zero that \(K_0(\mathcal{D}^{\mathrm{sg}}(X))\) is finite torsion, and that \(K_1(\mathcal{D}^{\mathrm{sg}}(X)) = 0\). One of the main applications is that algebraic varieties with isolated quotient singularities satisfy rational Poincaré duality on the level of the Grothendieck group; this allows computing the Grothendieck group of such varieties in terms of their resolution of singularities. Other applications concern the Grothendieck group of perfect complexes supported at a singular point and topological filtration on the Grothendieck groups. \(K\)-theory of singular varieties; quotient singularity; derived category; singularity category Applications of methods of algebraic \(K\)-theory in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Derived categories, triangulated categories, \(K\)-theory of schemes \(K\)-theory and the singularity category of quotient singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The main result of this paper is the proof a conjecture of Hausel and Rodriguez-Villegas on the cohomology of the moduli space of stable \(\mathrm{PGL}_n\)-Higgs bundles \(M^d_{\mathrm{PGL}_n}\), on a smooth projective complex curve \(C\), for any rank \(n\) and any degree \(d\) coprime to \(n\). Denote the canonical bundle of \(C\) by \(\Omega\).
The space \(M^d_{\mathrm{PGL}_n}\) is a quotient of the moduli space \(M_n^d\) of rank \(n\) and degree \(d\) \(\mathrm{GL}_n\)-Higgs bundles. The coprimality condition \((n,d)=1\) assures the smoothness of \(M_n^d\) and hence the fact that \(M^d_{\mathrm{PGL}_n}\) has only finite quotient singularities.
Consider its rational cohomology \(H^*(M^d_{\mathrm{PGL}_n})\). Hausel and Rodriguez-Villegas conjectured that the intersection form on \(H^*(M^d_{\mathrm{PGL}_n})\) vanishes identically (the case \(n=2\) has been shown to be true around twenty years ago). The main theorem of the paper proves this conjecture. An equivalent formulation is that the forgetful map from compactly supported cohomology \(H_c^*(M^d_{\mathrm{PGL}_n})\to H^*(M^d_{\mathrm{PGL}_n})\) is zero.
Since \(M^d_{\mathrm{PGL}_n}\) contracts to the nilpotent cone -- the fiber over zero of the Hitchin fibration \(h_{\mathrm{PGL}_n}:M^d_{\mathrm{PGL}_n}\to \bigoplus_{i=2}^nH^0(C,\Omega^{\otimes i})\) -- whose dimension is half that of \(M^d_{\mathrm{PGL}_n}\), it follows that \(M^d_{\mathrm{PGL}_n}\) has no cohomology of degree higher than middle, and hence by Poincaré duality the same holds for lower than middle degree compactly supported cohomology. Thus the theorem only requires proof for the middle degree cohomology group. It turns out that this group is freely generated by the cycle classes of the irreducible components of the nilpotent cone \(h_{\mathrm{PGL}_n}^{-1}(0)\). Moreover, these components are quotients by \(\mathrm{Pic}_C\) of the irreducible components of the nilpotent cone of \(M_n^d\), i.e., \(h^{-1}(0)\) where \(h:M_n^d\to \bigoplus_{i=1}^nH^0(C,\Omega^{\otimes i})\) is the Hitchin fibration on \(M_n^d\).
The way the author uses to study these components of \(h^{-1}(0)\) (except the one corresponding to the moduli of stable vector bundles) is deform them to \(h^{-1}(a)\), where \(a\in \bigoplus_{i=1}^nH^0(C,\Omega^{\otimes i})\) is a point in the Hitchin base, whose spectral curve is reducible and often non-reduced. From these explicit deformations, the author is able to express the cycle class of the irreducible component of \(h^{-1}(a)\) containing the deformed component in terms of the cycle classes of the irreducible components of \(h^{-1}(0)\). This is done via the \(\mathbb{C}^*\)-action on \(M_n^d\) scaling the Higgs field. The explicit form of these deformed components is enough to prove that the cup product between their classes is zero, and the same holds for the cup product between their classes and the class of the moduli of stable bundles of type \((n,d)\). Then, by the above mentioned explicit relation between the cycle classes, the same holds using the irreducible components of \(h^{-1}(0)\). Finally the same holds for the self-intersection of the class of the moduli of stable bundles, using the fact that its Euler characteristic is zero. This gives a brief overview of the proof of the main theorem.
Actually the irreducible components of \(h^{-1}(0)\) are in bijective correspondence to the \(\mathbb C^*\)-fixed point subvarieties. These subvarieties have a modular interpretation as the moduli space of holomorphic chains, whose semistability condition depends on a tuple of real parameters. The argument to complete the proof of the main theorem uses the fact that the moduli spaces of chains are irreducible (for stability parameters whose successive differences of its components are bigger than \(2g-2\)). This is proved in this paper as well, being the other main result, due to its clear independent interest. Higgs bundles; intersection form; moduli spaces of chains Jochen Heinloth, ''The intersection form on moduli spaces of twisted \(P G L_n\)-Higgs bundles vanishes'', Vector bundles on curves and their moduli, Stacks and moduli problems The intersection form on moduli spaces of twisted \(\mathrm{PGL}_n\)-Higgs bundles vanishes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 first part of this article, the classical study of binary sextics is used to describe the space of smooth curves of genus 2, \(M_2\), as an open subset of the weighted projective space \(X = \mathbb{P}(1,2,3,5)\). Further study of the birational geometry of \(\overline{M}_2\) and its divisor classes are then used to describe \(\overline{M}_2\) as a blow-up of \(X\). In the second part of the article, log canonical models of moduli spaces of stable curved are studied. The guideline problem is to understand the canonical model of \(\overline{M}_g\) when \(\overline{M}_g\) is of general type. Let \(V\) be a normal projective variety and \(D\) a \(\mathbb{Q}\)-divisor such that \(K_V+D\) is \(\mathbb{Q}\)-Cartier. The pair \((V,D)\) is called a strict log canonical model if \(K_V+D\) is ample, \((V,D)\) has log canonical singularities, and \(V-D\) has canonical singularities. It is proved that for \(g \geq 4\) the pair \((\overline{M}_g, \Delta)\) is a strict log canonical model where \(\Delta\) is the boundary divisor. When \(g=2\), \(K_{\overline{M}_2}+\Delta\) is not effective. To recover an analogous result, log canonical models of the moduli stack \(\overline{\mathcal M}_2\) with respect to \(K_{\overline{\mathcal M}_2} + \alpha \delta\) are studied for various values of \( 0 \leq \alpha \leq 1\). It is shown that this log canonical model is \(\overline M_2\) when \(9/11 < \alpha \leq 1\), and it is the invariant theory quotient \(X \simeq \mathbb{P}(1,2,3,5)\) when \(7/10 < \alpha \leq 9/11\). If \(\alpha < 7/10\), then the log canonical divisor is not effective. Hassett, B.: Classical and minimal models of the moduli space of curves of genus two. In: Geometric Methods in Algebra and Number Theory, Volume 235 of Progress in Mathematics, pp. 169-192. Birkhäuser, Boston (2005) Families, moduli of curves (algebraic), Minimal model program (Mori theory, extremal rays) Classical and minimal models of the moduli space of curves of genus two | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In the paper under review, the authors follow relations of the triality with theory of simple singularities in the sense of V. I. Arnold, S. M. Gusein-Zade, and A. N. Varchenko [\textit{V. I. Arnol'd} et al., Singularities of differentiable maps. Volume I: The classification of critical points, caustics and wave fronts. Transl. from the Russian by Ian Porteous, ed. by V. I. Arnol'd. Boston-Basel-Stuttgart: Birkhäuser (1985; Zbl 0554.58001)].
The authors recall É. Cartan's construction (from 1925) of the triality automorphism of the Lie algebra \(\mathfrak{so(8)}\), give a matrix representation for the real form \(\mathfrak{so(4,4)}\), describe how the cohomology homomorphism induced by the triality automorphism operates on the characteristic classes in the cohomology of the classifying space \(B\text{Spin}(8)\), and study the triality automorphism of the singularity \(D_4\).
The reviewer remarks that the following paper (not listed in the references) is closely related to the cohomology aspect of this work: [\textit{A. Gray} and \textit{P. Green}, Pac. J. Math. 34, 83--96 (1970; Zbl 0194.22804)]. triality; characteristic class; Lie algebra; singularities of smooth functions Homology of classifying spaces and characteristic classes in algebraic topology, Exceptional (super)algebras, Singularities of surfaces or higher-dimensional varieties, Local complex singularities Triality, characteristic classes, \(D_4\) and \(G_2\) singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper contains a thorough treatment of geometric, arithmetic and algorithmic aspects of hyperelliptic curves of small genus with special emphasis on genus \(3\) curves.
Shioda's computation of the graded ring \(\mathcal{I}_8\) of invariants of binary octics is reviewed and shown to be valid for any algebraically closed field of positive characteristic \(p>7\). The ring is generated by \(9\) fundamental invariants \(J_2,\dots,J_{10}\) which satisfy \(5\) relations. This yields a representation of the coarse moduli space of genus \(3\) hyperelliptic curves as a projective variety defined by the five Shioda relations on a weighted projective space of dimension \(9\) whose points are of the form \((J_2: J_3:\dots : J_{10})\) and the coordinates have weights \(2,3,\dots,10\). This description of the moduli space facilitates the enumeration of rational points in the strata determined by the automorphism group, over a finite field. The paper contains an exhaustive description of these strata and their characterization in terms of equations defined by the invariants, and it corrects several mistakes that occur in previous monographs on this topic.
The construction of curves with prefixed values of the fundamental invariants is also tackled. The authors are inspired by the general method of \textit{J.-F. Mestre} [Effective methods in algebraic geometry, Proc. Symp., Castiglioncello/Italy 1990, Prog. Math. 94, 313--334 (1991; Zbl 0752.14027)], based on the computation of the eight points of intersection of a non-singular conic \(\mathcal{Q}\) with a degree \(4\) curve \(\mathcal{H}\) whose coefficients are invariants. The construction depends on the stratum of the moduli space to which the point \((J_2: J_3:\dots : J_{10})\) belongs. Algorithms for the computation of the coefficients of \(\mathcal{H}\) as polynomials in the Shioda invariants are developed and specific tricks are used in the cases where the conic \(\mathcal{Q}\) is singular.
Finally, several arithmetic questions are addressed, again by working specifically on the different strata of the moduli space. Among others, the following problems are solved for each stratum: Is the field of moduli automatically a field of definition? Can the curve be always hyperelliptically defined over the field of moduli? Can one construct a model over the field of moduli when there is no obstruction?
A Magma code containing the various algorithms to check the computational assertions, calculate invariants and construct curves with prescribed invariants is available on the web page of the authors. automorphism; covariant; field of moduli; field of definition; genus 3; invariant; moduli space R.~J. Atkin and N.~Fox. \textit{An introduction to the theory of elasticity}. Longman, London, 1980. Longman Mathematical Texts. Computational aspects of algebraic curves, Actions of groups on commutative rings; invariant theory, Families, moduli of curves (algebraic), Automorphisms of curves Hyperelliptic curves and their invariants: geometric, arithmetic and algorithmic aspects | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 an explicit recipe for determining iterated local cohomology groups with support in ideals of
minors of a generic matrix in characteristic zero, expressing them as direct sums of indecomposable
D-modules. For nonsquare matrices these indecomposables are simple, but this is no longer true for
square matrices where the relevant indecomposables arise from the pole order filtration associated with
the determinant hypersurface. Theorem 1.1 determines the class in \(\Gamma_D\) of the local
cohomology groups of each \(D_p\), thus generalizing the main result of [\textit{C. Raicu} and \textit{J. Weyman}, Algebra Number Theory 8, No. 5, 1231--1257 (2014; Zbl 1303.13018)]
which addresses the case \(p = n\).
For nonsquare matrices, Theorem 1.1, together with the fact that \(\bmod_{\mathrm{GL}}(D_X )\) is semisimple, gives a description of Lyubeznik numbers. Specializing our results to a single iteration, they determine the Lyubeznik numbers for all generic determinantal rings and prove the vanishing of a range of local cohomology groups. Next, they use the quiver description of the
category \(\bmod_{\mathrm{GL}}(D_X )\) in conjunction with the vanishing results to provide an inductive proof of Theorem 1.6. determinantal varieties; local cohomology; Lyubeznik numbers; equivariant \(\mathcal{D}\)-modules Local cohomology and commutative rings, Homological functors on modules of commutative rings (Tor, Ext, etc.), Determinantal varieties Iterated local cohomology groups and Lyubeznik numbers for determinantal 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 In the paper under review, the authors introduce a new class of algebraic varieties called the frieze varieties. The frieze variety is defined in an elementary recursive way by constructing a set of points in the affine space. From a more conceptual viewpoint, the coordinates of these points are specializations of cluster variables in the cluster algebra associated to the quiver. Note that each frieze variety is determined by an acyclic quiver.
It is well known that the acyclic quiver is representation finite if and only if its underlying graph is a Dynkin diagram of type \(\mathbb{A}\), \(\mathbb{D}\) or \(\mathbb{E}\), and it is tame if and only if the underlying graph is an affine Dynkin diagram of type \(\widetilde{\mathbb{A}}\), \(\widetilde{\mathbb{D}}\) or \(\widetilde{\mathbb{E}}\). All other acyclic quivers are wild.
In this paper, the authors give a new characterization of the finite-tame-wild trichotomy for acyclic quivers in terms of their frieze varieties.
More precisely, they prove that an acyclic quiver is representation finite, tame, or wild, respectively, if and only if the dimension of its frieze
variety is \(0\), \(1\), or \(\geq 2\), respectively.
Finally, let us mention that there are several characterizations of the finite-tame-wild trichotomy in the literature, however, it seems that the characterization given by the authors is the first one in terms of numerical invariants that are integers. frieze variety; representation type of quivers; cluster algebra; Coxeter matrix; spectral radius Representation type (finite, tame, wild, etc.) of associative algebras, Cluster algebras, Representations of quivers and partially ordered sets, Special varieties Frieze varieties: a characterization of the finite-tame-wild trichotomy for acyclic 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 Resolution of singularities is one of the origins of algebraic geometry. There is a long way from Newton's method to determine branches of a plane curve, Puiseux-series', the work of M. Noether, Riemann, the geometers of the Italian school, and many others. In the middle of the 20th century, Zariski and Abhyankar prepared the ground to study the general case of arbitrary dimension which has been settled in characteristic 0 by \textit{H. Hironaka} [Ann. Math. (2) 79, 109--203, 205--326 (1964; Zbl 0122.38603)]. Hironaka's result (which had not been recognized in its full importance over the first years) is meanwhile one of the most famous, frequently used theorems of algebraic geometry, though apparently only few people have gone through all details of its demanding proof. Apart from the still unsolved problem of resolution in positive characteristics -- which motivates a closer look for alternative proofs of the characteristic 0 case -- there are other reasons for further studies: The resolution problem admits modifications, one of them due to \textit{A. J. de Jong} [in: Resolution of singularities. A research textbook in tribute to Oscar Zariski. Prog. Math. 181, 375--380 (2000; Zbl 1022.14005)], which led to a solution of the general problem up to alterations. On the other hand, appearance of computers in the offices of most mathematicians during the last decade of the 20th century has led to increasing interest in algorithmic questions. Refined resolution algorithms have been developed [cf. \textit{O. Villamayor}, in: Real analytic and algebraic geometry. Proc. int. conf. Trento 1992, 277--291 (1995; Zbl 0930.14039)], and there is a continued interest in better understanding the ideas of Hironaka's original proof [cf. \textit{H. Hauser}, Bull. Am. Math. Soc., New Ser. 40, No.3, 323--403 (2003; Zbl 1030.14007)]. Several more recent proofs include results of \textit{E. Bierstone} and \textit{P. D. Milman} [Invent. Math. 128, No.2, 207--302 (1997; Zbl 0896.14006)], \textit{S. Encinas} and \textit{O. Villamayor} [in: Resolution of singularities. A research textbook in tribute to Oscar Zariski. Prog. Math. 181, 147--227 (2000; Zbl 0969.14007) and Rev. Mat. Iberoam. 19, No.2, 339--353 (2003; Zbl 1073.14021)], \textit{T. T. Moh} [Commun. Algebra 20, No.11, 3207--3249 (1992; Zbl 0784.14008)], \textit{O. Villamayor} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 22, No.1, 1--32 (1989; Zbl 0675.14003), Ann. Sci. Éc. Norm. Supér., IV. Sér. 25, No.6, 629--677 (1992; Zbl 0782.14009)], \textit{J. Wlodarczyk} [J. Am. Math. Soc. 18, No.4, 779--822 (2005; Zbl 1084.14018)]. It should be noted, that there exist first approaches to use computer-algebra systems for performing the resolution process of given singularities (cf. the one by \textit{G. Bodnar, J. Schicho} [J. Symb. Comput. 30, No.4, 401--428 (2000; Zbl 1011.14005)] using Maple and another one by \textit{A. Frühbis-Krüger, G. Pfister} [Mitt. Dtsch. Math.-Ver. 13, No.2, 98--105 (2005; Zbl 1084.14036)] for Singular, respectively).
The book under review provides as well an introduction as advanced treatment of the resolution problem. Its modern presentation of meanwhile classical ideas interacts with recent research on the topic (cf. e.g. \textit{J. Kollar} [``Resolution of Singularities - Seattle Lecture'', preprint, \texttt{http://arXiv.org/abs/math/0508332}] and results by the author). After a short introduction, Chapter 2 defines basic notions of smoothness, non-singularity, resolution, normalization and local uniformization, followed by chapter 3, containing a discussion of embedded resolution for curve singularities. Chapter 4 starts constructing the blowing up of an ideal and gives the general notion of resolution. The fifth chapter studies resolution of surface singularities and their embedded resolution (again in characteristic 0). Chapter 6 gives a complete proof for resolution of singularities in arbitrary dimension and characteristic 0, based on the work of Encinas and Villamayor. Chapters 7 and 8 cover additional topics: Local uniformization and resolution of surfaces in positive characteristics (in a modern version of Zariski's original proof) and an introduction to valuation theory in algebraic geometry, together with the problem of local uniformization. An appendix contains technical material on the singular locus and semi-continuity-theorems used in the previous text. This book is pleasant to read and gives with its exercises a well prepared basis for a graduate course. ,\textit{Resolution of Singularities}, Graduate Studies in Mathematics, vol. 63, American Mathematical Society, Providence, RI, 2004.http://dx.doi.org/10.1090/gsm/063.MR2058431 Research exposition (monographs, survey articles) pertaining to algebraic geometry, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties 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 The main object of the paper under review is a linear algebraic group \(G\) (not necessarily connected) defined over a one-variable function field \(F\) over a complete discretely valued field \(K\). The group \(G\) is assumed \(F\)-rational (i.e., each connected component of \(G\) is an \(F\)-rational variety), thus including an important particular case where \(G\) is an orthogonal group. The focus is on local-global principles for torsors under \(G\). The authors give a necessary and sufficient condition for such groups to satisfy these local-global principles with respect to a finite set of overfields and compute the precise obstruction, which turns out to be finite. The obstruction is given in terms of the reduction graph \(\Gamma\) of a normal projective model \(\hat X\) of \(F\) over the valuation ring \(T\) of \(K\). Other characterizations are given in terms of split covers and split extensions, implying that the obstruction is independent of the choice of a regular model. The authors also consider the obstruction to a local-global principle with respect to discrete valuations. In particular, they show that if the residue field of \(T\) is algebraically closed of characteristic zero, this obstruction coincides with the previous one. Some of the above results carry over from torsors to more general homogeneous spaces.
Applications of these general results include local-global principles for various objects, such as the Witt index of a quadratic form, the Witt group of quadratic form classes, the index of a central simple algebra, and the splitting of such an algebra.
As in their earlier work, the methods rely on patching techniques. In the paper under review, this approach is developed further, and some of technical results are interesting in their own right, such as an exact sequence of Mayer-Vietoris type. linear algebraic group; torsor; function field; local-global principle; Galois cohomology David Harbater, Julia Hartmann, and Daniel Krashen. Local-global principles for torsors over arithmetic curves. Amer.\ J.\ Math., \textbf{137}(6) (2015), 1559--1612. DOI 10.1353/ajm.2015.0039; zbl 1348.11036; MR3432268; arxiv 1108.3323 Galois cohomology of linear algebraic groups, Algebraic functions and function fields in algebraic geometry, Rational points, Algebraic theory of quadratic forms; Witt groups and rings Local-global principles for torsors over arithmetic curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The sheaf of principal parts \(J^k(E)\) has been studied by many authors (DiRocco, Grothendieck, Laksov, Maakestad, Perkinson, Piene, Sommese etc). The sheaf \(J^k(E)\) of an \(O_X\)-module \(E\) where \(X\) is a scheme has a left and right structure as \(O_X\)-module and the left structure of \(J^k(E)\) has been studied by several authors in the case where \(X\) is projective \(n\)-space over an algebraically closed field of characteristic zero. The aim of the paper under review is to complete this study and to relate it to the theory of representations of quivers.
The projective space \(\mathbb{P}(V^*)\) may be realized as a quotient \(\mathrm{SL}(V)/P\) where \(P\) is a parabolic subgroup of \(\mathrm{SL}(V)\) and there is an equivalence of categories between the category of \(P\)-modules and the category of \(\mathrm{SL}(V)\)-linearized vector bundles on \(\mathbb{P}=\mathbb{P}(V^*)\). The category \(C\) of vector bundles on \(P\) with an \(\mathrm{SL}(V)\)-linearization is an abelian category, hence by Freyd's full embedding theorem it follows the category \(C\) is equivalent to a full subcategory of the category \(mod(A)\) of left modules on an associative ring \(A\). One aim of the paper is to describe the associative ring \(A\) associated to projective space \(P\) and to construct the \(A\)-module corresponding to the sheaf of principal parts \(J^k(E)\).
In section one of the paper the author gives the motivation for the writing of the paper and the main results of the paper.
In section two of the paper the author gives the definition of the sheaf of principal parts \(J^k(E)\) of any \(O_X\)-module \(E\) on any scheme \(X\) following the standard construction using the infinitesimal neighborhood of the diagonal and mentions some properties of this construction: He gives a description of the fiber of the principal parts, the fundamental exact sequences and the relationship with sheaves of differential operators.
In section three the author introduce the concepts of algebraic groups, homogeneous spaces and homogeneous vector bundles and mention the fact that if \(E\) is a \(G\)-linearized homogeneous vector bundle on a homogeneous space \(G/H\) it follows \(J^*(E)\) has a canonical \(G\)-linearization. The author mentions the notions of a Cartan decomposition of a parabolic Lie algebra, the notion of a maximal weight vector of a \(p\)-module where \(p\) is a parabolic Lie algebra and the notion of an irreducible homogeneous vector bundle. The author ends section three with an introduction to the notion of quiver representations. He also constructs the quiver \(Q_V\) associated to projective space \(\mathbb{P}=\mathbb{P}(V^*)\). He moreover gives an explicit construction of the equivalence between the category of representations of \(Q_V\) and the category of homogeneous vector bundles on \(P\).
In section four the author mentions known results on \(J^k(O(d))\) on \(P\) and introduces some notions defined in [\textit{D. Perkinson}, Compos. Math. 104, 27--39 (1996; Zbl 0895.14016)]. He uses these notions and some explicit formulas to prove the existence of a decomposition \(J^k(O(d))\cong Q_{k,d}\oplus J^d(O(d))\) where \(Q_{k,d}\) is an explicitly defined vector bundles on \(P\). The author defines a map
\[
n^{k-d}:S^k(V)\otimes O(d-k) \rightarrow S^k(V) \otimes O_P
\]
and an isomorphism \(Q_{k,d}\cong \ker(n^{k-d})\). He proves that the map \(n^{k-d}\) is an \(\mathrm{SL}(V)\)-invariant differential operator. The author moreover proves some properties of the bundles \(Q_{k,d}\) and use these properties to give an explicit construction of the \(Q_V\)-representation of \(J^k(O(d))\). The paper ends with a proof of the fact that the Taylor truncation map has maximal rank in the cases where \(h \leq k\).
Note: In Proposition 4.6 the author states the existence of an \(\mathrm{SL}(V)\)-equivariant isomorphism
\[
J^k(O(d)) \cong S^k(V) \otimes O(d-k)
\]
In other papers [the reviewer, Proc. Am. Math. Soc. 133, No. 2, 349--355 (2005; Zbl 1061.14040)] it was proved that \(J^k(O(d))\cong S^k(V^*)\otimes O(d-k)\). One may suspect there is an error in the paper since \(S^k(V)\) and \(S(V^*)\) are different as \(\mathrm{SL}(V)\)-modules. The difference between the author's paper and the reviewer's paper is that Re is considering \(\mathbb{P}(V)\) -- projective space parametrizing lines in \(V^* \) where Maakestad is considering \(\mathbb{P}(V^*)\) -- projective space parametrizing lines in \(V\). principal parts; quiver representation; stable; vector bundle; projective space Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Grassmannians, Schubert varieties, flag manifolds, Sheaves and cohomology of sections of holomorphic vector bundles, general results Principal parts bundles on projective spaces and quiver representations | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The complex Lie superalgebras \(\mathfrak{g}\) of type \(D(2,1;a)\) - also denoted by \(\mathfrak{osp}(4,2;a) \) -- are usually considered for ``non-singular'' values of the parameter \(a\), for which they are simple. In this paper we introduce five suitable integral forms of \(\mathfrak{g}\), that are well-defined at singular values too, giving rise to ``singular specializations'' that are no longer simple: this extends the family of simple objects of type \(D(2,1;a)\) in five different ways. The resulting five families coincide for general values of \(a\), but are different at ``singular'' ones: here they provide non-simple Lie superalgebras, whose structure we describe explicitly. We also perform the parallel construction for complex Lie supergroups and describe their singular specializations (or ``degenerations'') at singular values of \(a\). Although one may work with a single complex parameter \(a\), in order to stress the overall \(\mathfrak{S}_3\)-symmetry of the whole situation, we shall work (following Kaplansky) with a two-dimensional parameter \({\sigma} = (\sigma_1,\sigma_2,\sigma_3)\) ranging in the complex affine plane \(\sigma_1 + \sigma_2 + \sigma_3 = 0\). Lie superalgebras; Lie supergroups; singular degenerations; contractions Simple, semisimple, reductive (super)algebras, Deformations and infinitesimal methods in commutative ring theory, Noncommutative algebraic geometry Singular degenerations of Lie supergroups of type \(D(2,1;a)\) | 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 The concept of primitive form was introduced by \textit{K. Saito} to construct the period mapping for the universal deformation of a holomorphic function with isolated singularities. Since the primitive form is defined by a certain system of bilinear differential equations, the global existence of the primitive form is unknown except the cases of simple singularities und simple elliptic singularities.
The purpose of this paper is to construct the primitive forms associated with simple singularities as reduced symplectic 2-form on simultaneous resolution space. To this end, in \S 1, we give the symplectic geometric construction of Grothendieck's simultaneous resolutions of the adjoint quotients. In \S 2, we give the symplectic geometric construction of simultaneous resolutions of simple singularities (see theorem 2.6). Then we explain that the reduced symplectic 2-form on the simultaneous resolution space is just the primitive form associated with simple singularity (see corollary 2.7). This construction of the primitive forms is closely related to \textit{P. J. Slodowy}'s Lie-theoretic construction of the period mapping. symplectic quotients; simultaneous resolutions of singularities Yamada, H., Symplectic reduction and simultaneous resolution of simple singularities, submitted. Global theory and resolution of singularities (algebro-geometric aspects), Homogeneous spaces and generalizations, Simple, semisimple, reductive (super)algebras, Modifications; resolution of singularities (complex-analytic aspects) Symplectic quotients and simultaneous resolutions of simple singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The present article proves the threefold isolated singularity case of a famous conjecture relating two seemingly completely independent notions: log canonical and \(F\)-pure singularities.
To define \(F\)-pure singularities, we may work locally. An affine variety \(Y\) over a perfect field \(k'\) of characteristic \(p>0\) is \(F\)-pure if the \(\mathcal{O}_Y\)-homomorphism \(\mathcal{O}_Y \to F_* \mathcal{O}_Y\) splits, where \(F : Y \to Y\) is the absolute Frobenius morphism of \(Y\). A variety \(X\) over a field \(k\) of characteristic zero is called of \(F\)-pure type if for every model \(X_A\) of \(X\) over a finitely generated \(\mathbb{Z}\) algebra \(A\) in \(k\), for a dense set of prime ideals \(q \subseteq A\), the reduction \(X_{q}\) mod \(q\) is \(F\)-pure.
On the other hand, the definition of log canonical singularities in characteristic zero follows a different patter. If \(X\) is a variety over a characteristic zero field \(k\), and \(f : Z \to X\) is a log-resolution of singularities, then we may write uniquely \(K_Z \equiv_X E\) for some exceptional divisor \(E\). That is, the intersection of \(K_Z\) and \(E\) with every \(f\)-exceptional curve agrees. Then we say that \(X\) has log canonical singularities, if every coefficient of \(E\) is at least \(-1\).
The surprising conjecture relating the above two notions is that a singularity in characteristic zero is log canonical if and only if it is of \(F\)-pure type. The present article proves this conjecture for the isolated threefold case. The idea is to relate the action of the Forbenius on \(\mathcal{O}_{X_{q}}\) to the action on \(H^{\dim V}(V, \mathcal{O}_V)\), where \(V\) is a minimal stratum in a log resolution of the locus with discrepancy \(-1\). The isomorphism is via Hodge theory, using a previous article of the first author [\textit{O. Fujino}, J. Math. Sci., Tokyo 18, No. 3, 299--323 (2011; Zbl 1260.14006)]. log canonical singularities; \(F\)-pure singularities FT O.~Fujino and S.~Takagi, On the \(F\)-purity of isolated log canonical singularities, Compositio Math. \textbf 149 (2013), no. 9, 1495--1510. Singularities in algebraic geometry, Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Minimal model program (Mori theory, extremal rays) On the \(F\)-purity of isolated log canonical singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(\Sigma_g\) be a closed surface of genus \(g\). Let \(G\) be a finite group, and \(p: \Sigma_g \to S^2\) a \(G\)-covering, that is, a finite regular covering whose deck transformation group is isomorphic to \(G\). Let \(E\) and \(B\) be compact oriented manifolds of dimension \(4\) and \(2\), and \(f : E \to B\) a smooth surjective map. A triple \((f, E, B)\) is a fibered \(4\)-manifold of the \(G\)-covering \(p\) if it satisfies (i) \(\partial E = f^{-1}(\partial B)\), (ii) \(f : E \to B\) has finitely many critical values \(\{ b_l \}_{l=1}^{n}\) (the inverse image \(f^{-1} (b_l)\) is called a singular fiber), and the restriction of \(p\) to the complement of singular fibers is a smooth oriented \(\Sigma_g\)-bundle, (iii) the structure group of this \(\Sigma_g\)-bundle is contained in the centralizer \(C(p)\) of the deck transformation group of \(p\) in \(\mathrm{Diff}_+ \Sigma_g\), (iv) the natural \(G\)-action on the complement of singular fibers extends to a smooth action on \(E\).
In this paper, it is shown that, for a \(G\)-covering \(p : \Sigma_g \to S^2\) with at least three branch points, there is a function \(\sigma_{loc}\) from the set of singular fiber germs of fibered 4-manifolds of the \(G\)-covering \(p\) to \(\mathbb{Q}\), such that for any fibered \(4\)-manifolds \((f, E, B)\) of \(G\)-covering, the signature of \(E\) is described as the sum \(\mathrm{Sign}(E) = \sum_{l=1}^{n} \sigma_{loc} (f_l)\), where \(f_l\) is a singular fiber germs of \(f : E \to B\). The function \(\sigma_{loc}\) is explained in Theorem 1.1 (the main theorem of this paper) and proved in Section 3 and 4. For the symmetric mapping class group \(\mathcal{M}_g(p) = \pi_0 C(p)\), the cobounding function of the pull back of the Meyer cocyle is considered to describe the local signature for fibered \(4\)-manifolds of the \(G\)-covering \(p\) when the condition (iv) is ignored and \(\mathcal{M}_g(p)\) satisfies some condition in Section 5. When \(G\) is abelian, the generating set of \(\mathcal{M}_g(p)\) is given in Section 6. When \(G = \mathbb{Z}_d\), for a special type of singular fiber germs, a formula for \(\sigma_{loc}\) is given in Section 7. local signatures; mapping class groups; fibered 4-manifolds Topology of Euclidean 4-space, 4-manifolds, Structure of families (Picard-Lefschetz, monodromy, etc.) A local signature for fibered 4-manifolds with a finite group action | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities According to the author, the work described in this article was motivated by a desire to understand, from a general point of view, the results of Felix Klein on the equations defining modular curves of prime order. From this perspective, the article's primary conclusion is theorem 7.8 which states that for a prime \(p\geq 11\) and not equal to \(13\), the modular curve \(X(p)\) may be constructed geometrically (in the set theoretic sense) from a certain \(3\)-tensor, i.e., the curve is the intersection of the covariants of the \(3\)-tensor. Inspired by Klein's work on \(X(7)\) and \(X(11)\) and their relationship to the invariants and covariants of \(\text{PSL}_2({\mathbb{F}}_7)\) and \(\text{PSL}_2({\mathbb{F}}_{11})\) respectively, the author investigates the ring of invariants of the semi-direct product \(\text{SL}_2({\mathbb{F}}_q)\cdot \text{Aut}({\mathbb{F}}_q)\) (where \(q\) is a power of \(p\)) and the symplectic group \(\text{Sp}({\mathbb{F}}_p^r)\), acting on certain Weil representations. (The Weil representations are described in an extensive appendix.) For any weakly self-adjoint representation, the author describes a pairing on the ring of invariants. The ring of invariants along with this pairing is called the bicycle of invariants.
The number of invariants required to generate the bicycle of invariants is often fewer than the number required to generate the corresponding ring of invariants. In conjecture 4.2, the author describes a conjectured generating set for the bicycle of invariants of certain Weil representations of \(\text{SL}_2({\mathbb{F}}_q)\). Conjecture 5.2 leads to a conjectured generating set for the bicycle of invariants of even degree for certain Weil representations of \(\text{Sp}({\mathbb{F}}_p^r)\). Both conjectures constitute applications of the bicycle conjecture (Conjecture 2.4). Although the author acknowledges that the bicycle conjecture is not true as stated, he believes that a suitable refinement is true and that the conjecture does constitute a principle for identifying generators for a bicycle of invariants. ring of invariants; modular curve; Hessean; Weil representations; bicycle of invariants Geometric invariant theory, Actions of groups on commutative rings; invariant theory, Group actions on varieties or schemes (quotients), Automorphism groups of \(\mathbb{C}^n\) and affine manifolds, Vector and tensor algebra, theory of invariants Invariants of \(\text{SL}_2(\mathbb{F}_q)\cdot\text{Aut}(\mathbb{F}_q)\) acting on \(\mathbb{C}^n\) for \(q=2n\pm 1\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities \textit{Y.~Ruan} [Cohomology ring of crepant resolutions of orbifolds, preprint, math.AG/0108195] formulated the cohomological minimal model conjecture, which asserts that \(K\)-equivalent projective manifolds have isomorphic quantum corrected rings. The quantum corrected product corresponding to a birational map is defined similarly to the usual quantum product, with the difference that one takes into account only contributions coming from exceptional effective curves.
The authors answer in affirmative the conjecture in the case of simple Mukai flops:
\[
\mathbb P^n\cong Z\subset X \overset{\pi}{\dasharrow} X'\supset Z'\cong (\mathbb P^n)^\vee.
\]
In this case, the quantum corrected product is defined by
\[
\langle \alpha *_{\pi}\beta,\gamma \rangle=\sum_{d\geq 0} \Psi^X_{d\ell}(\alpha,\beta,\gamma)q^d,
\]
where \(\Psi^X_{d\ell}\) is the \(3\)-point Gromov-Witten invariant of \(X\), and \(\ell\in H_2(Z)\) is the class of a line. It can be written as \(\alpha *_\pi\beta=\alpha\cup\beta+\alpha *_{qc} \beta\).
For two projective varieties \(X\) and \(X'\) related by a simple Mukai flop, the authors give an explicit isomorphism between their cohomology rings. Secondly, they observe that the correction terms in the quantum corrected product depends only on the local geometry of the exceptional locus, and reduce the problem to computations on the moduli space of stable maps to \(\mathbb P^n\):
\[
\Psi^X_{d\ell}(\alpha,\beta,\gamma)=\int_{\overline{\mathcal M}_{0,3} (\mathbb P^n,d)}ev_1^*(\alpha)\cdot ev_2^*(\beta)\cdot ev_3^*(\gamma)\cdot\Phi,
\]
where the \(ev\)'s are the evaluation morphisms, and \(\Phi\) is an obstruction class corresponding to the embedding \(Z\subset X\). Using the localization formula for the virtual fundamental class of \textit{T.~Graber} and \textit{R.~Pandharipande} [Invent. Math. 135, 487--518 (1999; Zbl 0953.14035)], the authors prove that \(\Phi\) always contributes with a factor of zero, and therefore all invariants (of arbitrary genus) of the form above vanish. quantum corrected product; cohomological minimal model conjecture Lee Y P, Lin H W, Wang C L. Invariance of quantum rings under ordinary flops. ArXiv:math.AG/1109.5540 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Rational and birational maps, Minimal model program (Mori theory, extremal rays) Mukai flop and Ruan cohomology | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities \textit{C. Chevalley} [Tohoku Math. J. (2) 7, 14--66 (1955; Zbl 0066.01503)] provided a combinatorial construction of all simple affine algebraic groups over any field, which can be interpreted as providing a description of all simple affine groups as group schemes over the integers. The philosophy of Chevalley may be naturally generalized to supergeometry. One says that an `affine supergroup' is a representable functor from the category of commutative superalgebras to the category of groups. With R. Fioresi, the author constructed Chevalley supergroups whose tangent space at the identity is any classical simple Lie superalgebra different from \(D(2,1,a)\) [\textit{R. Fioresi} and \textit{F. Gavarini}, Mem. Am. Math. Soc. 1014, iii-v, 64 p. (2012; Zbl 1239.14045)]. The present paper fills in this gap. As the case of simple Lie superalgebras of Cartan type has also been solved by the author in [Forum Math. 26, No. 5, 1473--1564 (2014; Zbl 1320.14066)], this completes the program of constructing connected affine supergroups associated with any simple Lie superalgebra. simple Lie superalgebras; affine supergroups; representation of Lie superalgebras Gavarini, F., Chevalley supergroups of type {\(D(2,1;a)\)}, Proceedings of the Edinburgh Mathematical Society. Series II, 57, 2, 465-491, (2014) Supervarieties, Noncommutative algebraic geometry, Simple, semisimple, reductive (super)algebras Chevalley supergroups of type \(D(2,1;a)\) | 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 It was established by Boalch that Euler continuants arise as Lie group valued moment maps for a class of wild character varieties described as moduli spaces of points on \(\mathbb{P}^1\) by Sibuya. Furthermore, Boalch noticed that these varieties are multiplicative analogues of certain Nakajima quiver varieties originally introduced by Calabi, which are attached to the quiver \(\Gamma_n\) on two vertices and \(n\) equioriented arrows. In this article, we go a step further by unveiling that the Sibuya varieties can be understood using noncommutative quasi-Poisson geometry modelled on the quiver \(\Gamma_n\). We prove that the Poisson structure carried by these varieties is induced, via the Kontsevich-Rosenberg principle, by an explicit Hamiltonian double quasi-Poisson algebra defined at the level of the quiver \(\Gamma_n\) such that its noncommutative multiplicative moment map is given in terms of Euler continuants. This result generalises the Hamiltonian double quasi-Poisson algebra associated with the quiver \(\Gamma_1\) by Van den Bergh. Moreover, using the method of fusion, we prove that the Hamiltonian double quasi-Poisson algebra attached to \(\Gamma_n\) admits a factorisation in terms of \(n\) copies of the algebra attached to \(\Gamma_1\). Euler continuants; character varieties; Boalch algebra; nonommutative quasi-Poisson geometry; quasi-Poisson algebras Representations of quivers and partially ordered sets, Poisson algebras, Noncommutative algebraic geometry, Symplectic structures of moduli spaces, Momentum maps; symplectic reduction Euler continuants in noncommutative quasi-Poisson 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 There is a well-known relationship between periodic continued fractions and 2-dimensional cusp singularities. Let \(\pi\) : \(U\to V\) be the minimal resolution of a 2-dimensional cusp singularity (V,p). Then the exceptional set \(X=\pi^{-1}(p)\) is either a cycle of s rational curves with self-intersection numbers \(a_ 1,a_ 2,...,a_ s\leq -2\) at least one of which is strictly smaller than -2 (s\(\geq 2)\), or a rational curve with a node and with a self-intersection number \(a<0\). Then we can associate to it the periodic continued fraction
\(\omega =[[\overline{-a_ 1,-a_ 2,...,-a_ s}]]=(-a_ 1)-\underline 1| \overline{(-a_ 2)}-...-\underline 1/\overline{(-a_ s)}- \underline 1| \overline{(-a_ 1)}-...,\)
or
\(\omega =[[\overline{-a+2}]]=(-a+2)-\underline 1| \overline{(-a+2)}- \underline 1| \overline{(-a+2)}....\)
Conversely, we can construct a 2-dimensional cusp singularity and its resolution as above, from a periodic continued fraction \(\omega\) first by constructing a convex cone in \({\mathbb{R}}^ 2\) and then applying the theory of torus embeddings. Moreover, the dual graph of X can be thought of as a subdivision of a circle \(S^ 1\), with \(a_ 1,a_ 2,...,a_ n\) attached to s vertices as weights in this order. In this paper, we generalize the above relationship to higher dimensions and construct higher dimensional cusp singularities from suitable analogues of periodic continued fractions. The well-known Hilbert modular cusp singularities are special cases of the cusp singularities we obtain. cusp singularities; periodic continued fraction H.~Tsuchihashi 1983 Higher dimensional analogues of periodic continued fractions and cusp singularities \textit{Tohoku Math.~J.~(2)}35 4 607--639 Singularities in algebraic geometry, Continued fractions, Singularities of curves, local rings Higher dimensional analogues of periodic continued fractions and cusp singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This article contains a survey of results on ``generic singularities''. Many of the results discussed were obtained by the author. No proofs are presented.
More precisely, generic singularities are those that appear when we consider a sufficiently general linear projection \(\pi\) of a smooth \(r\)-dimensional projective variety \(X \subset {\mathbb P}^n\) into a linear subspace \( V \) (\( \approx {\mathbb P}^{m}\)) of \( {\mathbb P}^n\), where \( r+1 \leq m \leq 2r\), and (letting \(Y=\pi (X)\)) we exclude points of \(Y=\pi (X)\) in a suitable lower dimensional subvariety. Expanding results of M. Noether, E. Lluis, etc, J. Roberts developed the basic theory of these singularities in the 1970's, see \textit{J. Roberts} [Trans. Am. Math. Soc. 212, 229--268 (1975; Zbl 0314.14003)]. After briefly reviewing this theory, Zaare-Nahandi discusses some of his contributions. For instance, using the notation above, if \(y=\pi(x), x \in X\), is an analytically irreducible generic singularity, he has obtained a very explicit description of the induced homomorphism \(\pi ^* : {\hat {\mathcal O}}_{V,y} \to {\hat {\mathcal O}}_{X,x}\) and of the \textit{local defining ideal} of the singularity \(y\), namely Ker(\(\pi ^*\)). The local defining ideal is expressed in terms of minors of an associated matrix \({\mathcal M}\) with coefficients in \({\hat {\mathcal O}}_{V,y}\). The ring \({\hat {\mathcal O}}_{V,y}\) is isomorphic to a power series ring, and specializing some of the variables the matrix \(\mathcal M\) induces a matrix \({\mathcal M}_0\) with interesting properties. For instance, the defining ideal becomes a square-free monomial ideal. A simplicial complex may be associated to it, some of its properties are studied. As an application, a formula for the depth of \({\mathcal O}_{Y,y}\) is obtained and a partial answer to a conjecture of Andreotti, Bombieri and Holme weak normality of certain points of \(Y\) is gotten.
The author works over an algebraically closed field but if the characterisitc is positive the embedding \(X \subset {\mathbb P}^n\) must satisfy some extra conditions. generic projections; generic singularities; local defining ideal Singularities in algebraic geometry, Local theory in algebraic geometry, Projective techniques in algebraic geometry, Determinantal varieties, Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes Algebraic properties of generic singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this article, the authors prove that a compact complex manifold which has zero algebraic dimension and carries a holomorphic Cartan geometry of algebraic type, has infinite fundamental group.
Let $Y$ be a compact complex manifold. Recall that the \textit{algebraic dimension} of $Y$ is by definition the transcendental degree of its field of local meromorphic functions over the field $\mathbb{C}$. If $Y$ is projective algebraic then its algebraic and geometric dimensions coincide. Having zero algebraic dimension therefore implies being as far from the algebraic realm as possible. Moreover, since such manifolds do not accommodate any non-constant meromorphic function, they are neither approachable by the powerful methods of complex analysis (in several variables). This roughly explains why such manifolds are rather studied with geometric tools. Let $G$ be a connected complex Lie group and $H\subset G$ be its closed complex Lie subgroup. A \textit{holomorphic Cartan geometry of type $(G,H)$} on a complex manifold $X$ is some infinitesimal structure on $X$ modeled on the complex manifold $G/H$ (for a more precise definition and in particular for the meaning of ``algebraic type'' see Definitions 2.1, 2.2 and 2.3 in the article). The authors' main result is as follows: if $X$ is a compact complex manifold of algebraic dimension zero and $X$ carries a holomorphic Cartan geometry of algebraic type, then the fundamental group of $X$ is infinite (see Theorem 4.1 in the article).
This result is related to a sort of rigidity conjecture of Dumitrescu-McKay (the last two authors of the present article) asserting that compact complex manifolds which are simply connected and admit a holomorphic Cartan geometry of type $(G,H)$ are actually biholomorphic to $G/H$. It is perhaps worth mentioning here that it is known that if $S^6$ admits a complex structure $X$ then its algebraic dimension must be zero. Therefore the authors present result, among other things, implies that such a hypothetical $X$ cannot carry any holomorphic Cartan geometry of algebraic type. holomorphic Cartan geometry of algebraic type; algebraic dimension; fundamental group; algebraic type Other complex differential geometry, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Holomorphic bundles and generalizations Cartan Geometries on complex manifolds of algebraic dimension 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 In this paper, a generalization of the following statement in higher dimensions is considered:
Suppose that \(K\) is an algebraically closed field, and consider the set of couples \(\{(x_{i},y_{i}) \in K^2 : i=1,\dots d+1 \}\). Then, there exists a unique polynomial \(f\) of degree \(d\) such that \(f(x_{i})=y_{i}\). To be more precise, a degree \(d\) planar algebraic curve is uniquely determined by its intersection with \(d+1\) parallel lines \(\{x_{1}=a\}\).
Reformulating the statement in higher dimensions gives rise to the problem of reconstructing an algebraic variety from its intersection with parallel hyperplanes. One will consider this problem by proving in the first place two algebraic results and finally giving a result concerning the stated problem above. Let us introduce some terminology. For an ideal \(I\) in the ring of polynomials in \(n\) variables we call cross section the ideal \(I + <x_{1}-a>\).
The first result concerns the case when there exists finitely many known cross sections. The first part of the result states that the radical of \(I\) is equal to the intersection over all the \(a\) in \(K\) of the radical \(\mathrm{rad}(I + <x_{1}-a>)\). The second part states that: \(I=\bigcap_{a \in K}\bigcap_{k\geq 1} I+<x_{1}-a>^k.\)
The second result concerns the situation where there exist infinitely many known cross sections.
From the algebraic point of view, the problem is to recover the ideal \(I\) from the set \(\{(a, I+<x_{1}-a>):a\in S \}\), where \(S \subset K\) is infinite. Note that one can construct only varieties with no irreducible components included in a hyperplane \(\{x_{1}=a\}\). This condition is denoted by (1).
If this condition (1) is fulfilled then, \(\mathrm{rad}(I)=\bigcap_{a\in S} \mathrm{rad}(I+<x-a>)\), where \(S\) is an infinite set in \(K\) and \(I=\bigcap_{a\in S} I + <x_{1}-a >\). The third result discusses the possibility of reconstructing a variety from finitely many cross sections.
We will state this theorem:
Let \(I=<f_{1},\dots,f_{r}>\subset K[x_{1}, \dots,x_{n}]\) be an ideal, let \(S\) is a finite set in \(K\) and \(f \in K\) satisfy : \(\deg(f)\leq d\). Let \(\delta =\max\{\deg(f_{i}), i=1,\dots,r\}\). {\parindent=6mm \begin{itemize} \item[-] \(I\) satisfies condition (1) with respect to \(x_{1}\) and \(|S|> (d+1)\deg(V(I))\), then \(f\in \mathrm{rad}(I) \iff f\in \mathrm{rad}(I+<x_{1}-a>), \forall a \in S\) where \(\deg(V(I))\) is the maximum of the degrees of the irreducible components of \(V(I)\). \item [-] \(I\) satisfies condition (1) with respect to \(x_{1}\), and \(|S|> ( (d+2(\delta r)^{2^{n-1}} )^n +1)max\{d,\delta\}\) then \(f\in I \iff f\in I +<x_{1}-a> \forall a\in S .\)
\end{itemize}} polynomial interpolation; ideals; complexity Computational aspects of higher-dimensional varieties, Computational aspects and applications of commutative rings Interpolation of ideals | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The authors study the null-cone of a semi-simple algebraic group acting on a number of copies of its Lie algebra via the diagonal adjoint action.
Their interest in the question is motivated by applications in the study of ordinary deformations of Galois representations. This point of view is due to Snowden who studied the case \(\mathbf{sl}_2\) in [\textit{A. Snowden}, ``Singularities of ordinary deformation rings'', Preprint, \url{arXiv:1111.3654}]. In that case he shows that the null-cones are Cohen-Macaulay but not Gorenstein. In the case of characteristic zero the Snowden method amounts to proving that the null-cone has rational singularities.
In this paper the authors show that for \(\mathbf{sl}_3\) the null-cones do still have rational singularities, and hence are Cohen-Macaulay.
Moreover, they also show that this fails in general. For example, they show that the null-cone is not normal when the group is of type B2. In type A5 they further show that the normalization of the null-cone does not have rational singularities. They do this by giving estimates on cohomology groups of equvariant vector bundles on flag varieties. null-cone; linear groups; lie algebras K. Vilonen and T. Xue, The null-cone and cohomology of vector bundles on flag manifolds, Represent. Th. 20 (2016), 482--498. Homogeneous spaces and generalizations, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Linear algebraic groups and related topics The null-cone and cohomology of vector bundles on flag varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper is devoted to degenerations of bundle moduli. Over a family of complete curves of genus \(g\), which gives the degeneration of a smooth curve into one with nodal singularities, the authors build a moduli space which is the moduli space of holomorphic \(SL(n,\mathbb{C})\) bundles over the generic smooth curve in the family, and is a moduli space of bundles equipped with extra structure at the nodes for the nodal curves in the family. This moduli space is a quotient by \((\mathbb{C}^*)^s\) of a moduli space on the desingularization. Taking a maximal degeneration of the curve into a nodal curve built from the glueing of three-pointed spheres, the authors obtain a degeneration of the moduli space of bundles into a \((\mathbb{C}^*)^{(3g-3)(n-1)}\)-quotient of a \((2g-2)\)-th power of a space associated to the three-pointed sphere. Via the Narasimhan-Seshadri theorem [\textit{M. S. Narasimhan} and \textit{C. S. Seshadri}, Ann. Math. (2) 82, 540--567 (1965; Zbl 0171.04803)], the moduli space of \(SL(n,\mathbb{C})\) bundles on the smooth curve is a space of representations of the fundamental group into \(SU(n)\) (the symplectic picture). The authors obtain the degenerations also in this symplectic context, in a way that is compatible with the holomorphic degeneration, so that their limit space is also a \((S^1)^{(3g-3)(n-1)}\) symplectic quotient of a \((2g-2)\)-th power of a space associated to the three-pointed sphere.
This paper is organized as follows: Section 1 is an introduction to the subject. Sections 2 is devoted to symplectic degeneration. The authors begin in this section with the symplectic category of representations of the fundamental group of the Riemann surface. Representing the degenerations of the curve on an open set as a family of quadrics in the plane, the degeneration is easy to obtain, in terms of flat connections, as the restriction of a singular flat connection in the plane. This gives them by the way another desired property, that isomonodromic deformations should give them a holomorphic family, well-behaved in the limit. As a limiting flat connection, the authors obtain on the twice punctured surface constructed by unglueing the curve a connection with regular singular points at the punctures, with opposite residues. The reduction is a (partial) identification of the fibers at the singular points. Such a process has already been discussed earlier in [\textit{J. Hurtubise} et al., Am. J. Math. 128, No. 1, 167--214 (2006; Zbl 1096.53045); Ann. Global Anal. Geom. 28, No. 4, 351--370 (2005; Zbl 1095.14034)].
Section 3 deals with holomorphic models for parabolic moduli. In this section the authors give a holomorphic interpretation of all this. Referring to the above references, they deform the holomorphic bundles into bundles over the desingularized nodal curve equipped with a framed parabolic structure. One product of this by-discussion is an emphasis on a particularly apposite interpretation of parabolic weights, as the decay rates of sections of a connection with a regular singular point; alternately, as the eigenvalues of the residue of the connection at a singular point. They then turn to moduli; this is a fibered problem, over the family \(X_t\) of curves degenerating to a nodal curve \(X_0\). From the symplectic point of view, this is quite straightforward; one is simply dealing with a family of representations of the fundamental group, with some extra structure at \(t=0\). This already gives the family of moduli spaces as a topological space; the authors concentrate on the holomorphic structure. Fiberwise, this is given by the natural Narasimhan-Seshadri correspondence, with a variant introduced in [loc. cit.] for \(t=0\). The paper closes in Sections 4 with a discussion of multiple degenerations, allowing occurrence of several nodes. stable vector bundles; algebraic curves; nodal degenerations; isomonodromy; trinion Real algebraic and real-analytic geometry, Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills), Kähler manifolds Degenerations of bundle moduli | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper under review gives an analytic proof of the celebrated Birkar-Cascini-Hacon-McKernan theorem on the finite generation of the canonical ring of a smooth projective variety [\textit{C. Birkar, P. Cascini, C. D. Hacon} and \textit{J. McKernan}, J. Am. Math. Soc. 23, No. 2, 405--468 (2010; Zbl 1210.14019)]. The first analytic proof of finite generation theorem was given by \textit{Y.-T. Siu} [``A general non-vanishing theorem and an analytic proof of the finite generation of the canonical ring'', \url{arXiv:math/0610740}]. Although both Siu's and the author's approaches are analytic in nature, they are actually quite different. In spirit, Păun's proof is more close to the algebraic ones given by Birkar et al. and \textit{V. Lazić} [``Adjoint rings are finitely generated'', \url{arXiv:0905.2707}]. Besides using \textit{Shokurov polytopes technique} and adopting other ideas from Birkar et al. and Lazić's papers, the author heavily utilizes extension theorems previously established by himself [``Relative critical exponents, non-vanishing and metrics with minimal singularities'', \url{arXiv: 0807.3109}]. The flexibility of the analytic approach motivates most of the presentation. His method will probably be useful in other contexts as well.
Denote \({\mathcal E}_{Y,A}:=\{ \tau \in [0, 1]^N : K_X+Y_{\tau}+A \in \text{Psef}(X)\}\), where \(\text{Psef}(X)\) is the pseudo-effective cone of a nonsingular projective manifold \(X\), \((Y_j)_{j=1, \dots, N}\) is a set of nonsingular hypersurfaces of \(X\) with strictly normal crossings, \(A\) is an ample \(\mathbb{Q}\)-bundle on \(X\) and \(Y_{\tau}:=\sum_{j=1}^{N} \tau^j Y_j\). Let \(l^j: \mathbb{R}^r \rightarrow \mathbb{R}\), \(j=1, \dots, N\) be a set of affine forms defined over \(\mathbb{Q}\) such that \(0 \leq l^j(\theta) \leq 1-\varepsilon_0\) for all \(\theta \in {\mathcal L}\), where \({\mathcal L} \subset \mathbb{R}^r\) is a \(r\)-dimensional rational polytope and \(\varepsilon_0\) is a positive real number. Let \(d:=(d^0, \dots, d^r)\) be an element of \(\mathbb{Z}^{r+1}\) with divisible enough coordinates. Let \(\Gamma_d :=\{(m, \theta)\in \mathbb{Z}_{+} \times {\mathcal L}: \forall j=0, \dots, r, m \theta^j \in d^j \mathbb{Z}, l(\theta) \in {\mathcal E}_{Y,A}\}\). Denote \({\mathcal A}_r (X)\) the ring of holomorphic sections \(\bigoplus_{(m,\theta) \in \Gamma_d} H^0 \big (X, m(K_X+\sum_{j=1}^{N} l^j (\theta) Y_j +A ) \big )\). In this paper, the author proves the following result, which has been confirmed affirmatively by Birkar, Cascini, Hacon and McKernan in the framework of the Minimal Model Program.
Theorem. With notations as above, the following are true.
(1) The set \({\mathcal E}_{Y,A}\) is a rational polytope;
(2) The ring \({\mathcal A}_r (X)\) is finitely generated.
The structure of the paper is very simple. In section \(0\), the author introduces the main theorem and briefly explains the relation between his approach and other algebraic/analytic proofs of finite generation theorem. The core technical tool in his proof is the theory of closed positive currents. In section \(1\), the author introduces some basic definitions and notations. The most important ones are \textit{current with minimal singularities in the sense of Demailly} and \textit{current with minimal singularities in the sense of Siu}. He also mentions the translation technique previously used by \textit{C. Hacon} and \textit{J. McKernan} [``On the existence of flips'', \url{arXiv:math/0507597}], which ensures that the coefficients of \(Y_{\tau}\) lie in the interval \([0, 1-\varepsilon_0 ]\).
The proof of the rationality of \({\mathcal E}_{Y,A}\) is given in section \(2\). By the classical theory of convex sets, to show the rationality of \({\mathcal E}_{Y,A}\) is equivalent to prove that the set of extremal points of \({\mathcal E}_{Y,A}\) is isolated (Claim 2.0 in the paper). For an extremal point \(\tau_0 \in {\mathcal E}_{Y,A}\), the author shows its isolation in two separated cases: the numerical dimension \(nd(\{K_X+Y_{\tau_0}+A\})=0\), and \(nd(\{K_X+Y_{\tau_0}+A\})\geq 1\). The proof in the case \(nd(\{K_X+Y_{\tau_0}+A\})=0\) is pretty straightforward. For any \(\tau_k \in {\mathcal E}_{Y,A}\) approaching \(\tau_0\), the author constructs a current \(T_{k,\eta} \in \{K_X+Y_{\tau_{k,\eta}}+A\}\) such that \(0 \leq \tau_{k,\eta}=\tau_k +\eta (\tau_k-\tau_0) \leq 1\) and \(T_{k,\eta}\) is positive. This implies the isolation of \(\tau_0\) immediately because \(\tau_k\) is a convex combination of two points \(\tau_0\) and \(\tau_{k,\eta}\) in \({\mathcal E}_{Y,A}\) and neither of them is equal to \(\tau_k\). In the case \(nd(\{K_X+Y_{\tau_0}+A\}) \geq 1\), the proof is much more involved.
In order to show that the sequence \((\tau_k)\) cannot be extremal points of \({\mathcal E}_{Y,A}\), it is sufficient to prove that \(\tau_k\) belongs to the interior of the segment \([\tau_0, \tau_{k_0}]\) for some \(\tau_{k_0} \in {\mathcal E}_{Y,A}\) when \(k\) is large enough. To this purpose, the author creates a center \(S\) adapted to \((\tau_k)\) on a modification of \(X\) and constructs a set \({\mathcal E}_{|S}\) satisfying certain conditions. He then shows that the set \({\mathcal E}_{|S}\) is a rational polytope. Using induction on dimension, non-vanishing and extension theorems, and the relation between current with minimal singularities in the sense of Siu and that in the sense of Demailly, the author proves that \((\tau_k,a^0_{\min} (\tau_k), \rho_{\min} (\tau_k)) \in {\mathcal E}_{|S}\) for all \(k \in \mathbb{Z}_{+}\). Using Diophantine approximation and extension argument, the author shows that \((\tau_k,a^0_{\min} (\tau_k), \rho_{\min} (\tau_k))\) is a non-trivial convex combination of \((\tau_0,a_{0}, \rho_{0})\) and \((\tau'_{k_0},a'_{k_0}, \rho'_{k_0})\), where \((\tau'_{k_0},a'_{k_0}, \rho'_{k_0}) \in {\mathcal E}_{|S}\) when \(k \gg 0\). This implies that \(\tau_k\) is a non-trivial convex combination of \(\tau_0\) and \(\tau'_{k_0}\) in \({\mathcal E}_{Y,A}\). So Claim \(2.0\) and hence the rationality of the polytope \({\mathcal E}_{Y,A}\) follows.
The proof of the finite generation of \({\mathcal A}_r (X)\) is given in section \(3\). Due to the fact that the decomposition of the polytope \({\mathcal L}\) into simplexes induce a decomposition of the algebra \({\mathcal A}_r (X)\) as a direct sum, it is sufficient to deal with the case that \({\mathcal L}\) is a \(r\)-dimensional simplex. Suppose that \(\xi_1, \dots, \xi_{r+1}\) are vertices generating \({\mathcal L}\). Using non-vanishing theorem, the author first obtains non-zero section \(s_k \in H^0(X, m_0(K_X+\sum_{j=1}^{N} l^j(\xi_k) Y_j+A))\) for \(k=1, \dots, r+1\). In Proposition 3.1, the author shows that if there exists a finite set of sections \(\sigma_l \in H^0(X, m_l(K_X+\sum_{j=1}^{N} l^j (\theta_l) Y_j+A))\) satisfying certain conditions, then the algebra \({\mathcal A}_r (X)\) is finitely generated by \((\xi_k)\), \((\sigma_l)\), together with a finite number of pluricanonical sections of small degree. The rest of section \(3\) is devoted to the construction of \(\sigma_l\). We refer the interested reader to the original paper for technical details.
This paper is well-written. The main ideas and techniques of the proof are clearly presented. It is interesting to see another approach to a famous result in higher-dimensional algebraic geometry. A minor problem one may have in reading this paper is that some equations are labeled incorrectly (e.g. in the proof of Proposition 2.3.5 on page 385, ``By the relation (46)'' actually refers to the relation (24)). Luckily, the author has fixed this problem and posted an enlarged version of the paper at \url{hal.archives-ouvertes.fr/docs/00/45/23/74/PDF/2009-37.pdf}. finite generation theorem; rational polytope; pseudo-effective divisors; closed positive currents; minimal singularities; Shokurov polytopes technique; extension theorems Research exposition (monographs, survey articles) pertaining to algebraic geometry, Divisors, linear systems, invertible sheaves, Minimal model program (Mori theory, extremal rays), Polytopes and polyhedra Lecture notes on rational polytopes and finite generation | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let (V,p) be an isolated 2-dimensional hypersurface singularity. An isolated singularity is a cone over its link L. In dimension 2, L is a compact real 3-manifold whose oriented homeomorphism type determines and is determined by the weighted dual graph of a canonically determined resolution \(\pi\) : (M,A)\(\to (V,p)\). In this paper, we ask: What conditions does the existence of a hypersurface representative (V,p) put on a weighted dual graph?
Since a hypersurface singularity is Gorenstein, there exists an integral cycle K on the exceptional set A, i.e., its weighted dual graph, which satisfies the adjunction formula: for all irreducible components \(A_ i\) of the A, \(A_ i\cdot K=-A_ i\cdot A_ i+2g_ i-2\) where \(g_ i\) denotes the genus of \(A_ i\) as a Riemann surface.
Theorem. Let \(D=d_ 1A_ 1+d_ 2A_ 2+...+d_ nA_ n\) be the maximal ideal cycle which is a unique cycle with \(d_ i>0\) such that \(\pi^*({\mathcal M})/{\mathcal O}(-D)\) is supported at only a finite number of points. Suppose that for each \(A_ j\) which is exceptional of the first kind, \(A_ j\cdot D<0\). Then for all \(A_ i\), \(-k_ i\geq 1+(\nu - 3)d_ i\) holds with the multiplicities \(\nu\) of (V,p).
Theorem. Let (V,p) be as above with multiplicity \(\nu\) greater than or equal to 4. Then, the following inequality holds: \(-K\cdot K\geq 2+ \nu (\nu -2)(\nu -4),\) for a canonical divisor K. isolated hypersurface singularity; resolution of a singularity; multiplicity of a singularity; maximal divisor; canonical divisor -, The multiplicity of isolated two-dimensional hypersurface singularities , Trans. Amer. Math. Soc. 302 (1987), 489-496. Complex singularities, Modifications; resolution of singularities (complex-analytic aspects), Singularities in algebraic geometry, Local complex singularities The multiplicity of isolated two-dimensional hypersurface singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The aim of this paper is to study the monodromy of the Hitchin fibration for moduli spaces of parabolic \(G\)-Higgs bundles in the cases when \(G=SL(2,\mathbb{R})\), \(GL(2,\mathbb{R})\) and \(PGL(2,\mathbb{R})\). A calculation of the orbits of the monodromy with \(\mathbb{Z}_2\)-coefficients provides an exact count of the components of the moduli spaces for these groups. This paper is organized as follows : The first section is an introduction to the subject and summarizes the main results. In the second section the authors introduce terminology for the moduli spaces of parabolic \(G\)-Higgs bundles that they are primarily interested in this paper. In section 3, they consider the Hitchin fibration and the construction of the spectral curve for rank \(2\) parabolic Hitchin systems. They also discuss here the parabolic version of the BNR correspondence with particular focus on the subvarieties of the Picard group restricted to which the correspondence is one-to-one, as well as on the Prym variety of the spectral covering of the \(V\)-surface. Moduli spaces of parabolic (or non-parabolic) \(G\)-Higgs bundles can be decomposed into closed subvarieties, yet not necessarily connected components, for fixed values of appropriate topological invariants. In this section the authors describe such topological invariants for the moduli spaces they are interested in, namely for the rank 2 cases \(G=SL(2,\mathbb{R})\), \(GL(2,\mathbb{R})\) and \(PGL(2,\mathbb{R})\). Here, the authors include the discussion for the topological invariants leading to the minimum number of connected components, while in section 5 they study the monodromy action on \(2\)-torsion points on the lattices. Finally, section 6 and section 7 include the exact calculation of the number of orbits of the monodromy action. The paper is supported by an appendix concerning BNR correspondence for orbicurves. monodromy; parabolic Higgs bundle; Hitchin system Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Generalized geometries (à la Hitchin), Relationships between algebraic curves and integrable systems, Vector bundles on curves and their moduli Monodromy of rank 2 parabolic Hitchin 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 [For the entire collection see Zbl 0607.00005.]
A theorem of Mather-Yau showed that an isolated hypersurface singularity (at 0 of \({\mathbb{C}}^ n)\) is determined up to biholomorphic equivalence by the associated moduli algebra A. Namely, if f is a representative germ defining the singularity, then the moduli algebra is the local (commutative) Artinian algebra (over \({\mathbb{C}}):\) \({\mathbb{C}}\{z_ 1,...,z_ n\}/(f,\partial_ 1f,...,\partial_ nf)\), \(\partial_ i=\partial /\partial z_ i\). A special case of the theorem of Mather-Yau was obtained by Benson using a different method at about the same time.
A natural algebraic problem is to ''recognize'' the moduli algebras among all the local Artinian algebras (over \({\mathbb{C}})\)- the theorem of Mather- Yau fails for fields of positive characteristics as well as for the field of real numbers. To this end, Yau has conjectured that the derivation algebra L(V) of the moduli algebra A(V) is always solvable. It appears that this conjecture has been verified when \(n\leq 5\). A second natural question is to extract information from the moduli algebra that depend only on the topological type of the singularity (V,0).
Chapter 1 of the present work supplies the details of a result announced by the second author in previous works. Chapter 2 studies Kac-Moody Lie algebras arising from hypersurface singularities and looks into a number of examples. Chapter 3 describes the use of a computer in the calculation of singularity invariants. Such computations are likely to be useful in formulating as well as checking plausible conjectures. isolated hypersurface singularity; local Artinian algebras; derivation algebra; moduli algebra; solvable; Kac-Moody Lie algebras; invariants; computations Cohomology of Lie (super)algebras, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Singularities in algebraic geometry, Software, source code, etc. for problems pertaining to nonassociative rings and algebras, Solvable, nilpotent (super)algebras Lie algebras and their representations arising from isolated singularities: Computer method in calculating the Lie algebras and their cohomology | 0 |
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