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The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper continues the recent investigation of the role played by the determinants of conformal block vector bundles with regard to the birational geometry of the moduli space curves (genus zero, primarily, in the present case). These vector bundles were studied quite actively in the 90s and early 2000s due to their fascinating interplay between algebraic geometry (generalized theta functions), representation theory (Kac-Moody algebras and loop groups) and mathematical physics (conformal field theory and other variants). One of the high points of the theory from those days is the Verlinde formula, a remarkable formula involving various trigonometric functions, among other things, that computes the rank of these vector bundles.
The recent explosion of interest in this subject (cf. the references in the present paper, as well as some more developments since its publication) has stemmed entirely from a brilliantly insightful paper of \textit{N. Fakhruddin} [Contemp. Math. 564, 145--176 (2012; Zbl 1244.14007)], which
(i) observes that these bundles are all globally generated in genus zero, and hence induce morphisms from \(\overline{M}_{0,n}\) to Grassmannians, and
(ii) produces a recursive formula for the first Chern class of these bundles.
The relevance is that the morphisms to Grassmannians, when composed with the Plücker embedding, yield morphisms to projective space and hence basepoint free linear systems on \(\overline{M}_{0,n}\), namely, the complete linear systems associated to the first Chern classes (= divisor classes of the determinant bundles) computed by the recursive formula. Two natural questions then arise, which are explored in certain specific cases in the present paper:
(1) what portion of the nef cone of \(\overline{M}_{0,n}\) do these conformal block divisor classes span, and
(2) what is the (modular) geometry of the morphism they induce.
The present paper focuses on the conformal block bundles constructed from \(\mathfrak{sl}_2\) by varying the level and associating the defining representation to each marked point (so these divisor classes are, in particular, naturally \(S_n\)-invariant). The reason for restricting attention to these classes is that Fakhruddin [loc. cit.] (and later myself, ``Conformal blocks and rational normal curves'', \url{arXiv:1012.4835}]) studied fairly extensively the case of level one classes, and also because these particular higher-level classes admit some truly remarkable properties, which are the subject, and main results, of this paper. For instance, one of these divisors induces the natural morphism
\[
\overline{M}_{0,2g+2} \rightarrow \overline{M}_{0,2g+2}/S_{2g+2} \cong \overline{H}_g \subset \overline{M}_g \rightarrow \overline{A}_g^{\mathrm{Sat}}
\]
where \(H_g\) denotes the hyperelliptic locus and the compactification of the moduli space of principally polarized abelian varieties here is that of Satake. Another divisor class, via the same inclusion as the hyperelliptic locus, induces the morphism from \(\overline{M}_g\) to Schubert's pseudo-stable compactification when \(g=2\), and when \(3 \leq g \leq 11\) it induces a similar morphism but the image is currently unknown (!) and would be interesting to describe in modular terms; finally, for \(g \geq 12\) the present paper computes the nef divisor class on \(\overline{M}_g\) but we do not know whether a multiple of it is basepoint free (although by a result of \textit{S. Keel} [Commun. Algebra 31, No. 8, 3955--3982 (2003; Zbl 1051.14017)] we DO know this over a finite field!). This is a fascinating connection between conformal blocks in genus zero and modular birational geometry in positive genus. Yet another of the divisor classes studied in this paper induce natural morphisms to a GIT quotient parameterizing \(n\)-pointed plane conics introduced by myself and Simpson.
Other results in this paper include numerical properties of F-curves (boundary 1-strata) and their relation to these divisor classes, and a proof that these classes form a basis for the \(S_n\)-invariant Picard group, among other things. Since its publication there have been more steps towards understanding the geometry and geography of genus zero conformal block divisors, yet many questions (along the lines of those explored here) remain! moduli of curves; conformal blocks; nef cone; hyperelliptic locus V. Alexeev, A. Gibney, D. Swinarski, Higher-level sl_{2} conformal blocks divisors on \textit{M}_{0,\textit{n}}\textit{Proc. Edinb. Math. Soc}. (2) 57 (2014), 7-30. MR3165010 Zbl 1285.14012 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Minimal model program (Mori theory, extremal rays), Fine and coarse moduli spaces, Two-dimensional field theories, conformal field theories, etc. in quantum mechanics Higher-level \(\mathfrak{sl}_2\) conformal blocks divisors on \(\overline{M}_{0,n}\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper deals with finite-dimensional algebras \(\Sigma\) over an algebraically closed field \(k\) whose derived category \({\mathcal D}^b(\text{mod }\Sigma)\) of the category \(\text{mod }\Sigma\) of finite-dimensional modules over \(\Sigma\) is equivalent to the derived category \({\mathcal D}^b(\text{coh }\mathbb{X})\) of the category of coherent sheaves on a weighted projective line \(\mathbb{X}\) in the sense of \textit{W. Geigle} and \textit{H. Lenzing} [in: Singularities, representations of algebras, and vector bundles, Lect. Notes Math. 1273, 265-297 (1987; Zbl 0651.14006)].
To an almost concealed-canonical algebra, which by definition is an algebra of the form \(\text{End}(T)\), where \(T\) is a tilting sheaf in \(\text{coh }\mathbb{X}\) [see \textit{H. Lenzing} and the reviewer in: Representation theory of algebras, CMS Conf. Proc. 18, 455-473 (1996; Zbl 0863.16013)], there is associated the Tits quiver whose vertices are the indecomposable direct summands \(P_m\) of \(T\). Moreover, the Tits quiver has integer-valued arrows \(P_m@>-d_{m,l}>>P_l\) where \(d_{m,l}\) denotes the Euler form \(\langle S_l,S_m\rangle\) applied to the corresponding simple \(\Sigma\)-modules.
The author shows that the rank function \(rk\colon K_0(\mathbb{X})\to\mathbb{Z}\) satisfies an additive property which is expressed in terms of the bigraph associated to the Tits quiver. Further, the author proves that the result extends to the representation-infinite quasitilted algebras of canonical type in the sense of \textit{H. Lenzing} and \textit{A. Skowroński} [Colloq. Math. 71, No. 2, 161-181 (1996; Zbl 0870.16007)]. Finally, it is shown that the rank additivity does not generalize to the case of algebras \(\Sigma\) which are given by tilting complexes in \({\mathcal D}^b(\text{coh }\mathbb{X})\). finite-dimensional algebras; derived categories; finite-dimensional modules; categories of coherent sheaves; weighted projective lines; almost concealed-canonical algebras; tilting sheaves; Tits quivers; indecomposable direct summands; Euler forms; rank functions; representation-infinite quasitilted algebras; rank additivity; tilting complexes Thomas Hübner, Rank additivity for quasi-tilted algebras of canonical type, Colloq. Math. 75 (1998), no. 2, 183 -- 193. Representations of quivers and partially ordered sets, Representation type (finite, tame, wild, etc.) of associative algebras, Derived categories, triangulated categories, Vector bundles on curves and their moduli Rank additivity for quasi-tilted algebras of canonical type | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(K\) be an algebraically closed field of characteristic zero, \(V\) a finite-dimensional vector space over \(K\) endowed with a nondegenerate 2-form \(\omega\in\Lambda^2(V^*)\) and a finite subgroup \(\Gamma \subset\mathrm{Sp}(V)\). One assumes that there exists a resolution of the singularities of the quotient variety \(\pi: X\to V/\Gamma\), such that the symplectic form induced on the smooth part of \(V/\Gamma\) has an extension to a nondegenerate closed 2-form \(\Omega\in H^0(\Omega^2_X)\).
The main result of this paper is the following: There exists an equivalence of \(\mathcal{O}_V^\Gamma\)-linear triangulated categories of coherent sheaves \(D^b(\text{Coh}(X))\simeq D^b(\text{Coh}^\Gamma(V))\). The proof is based on the reduction to positive characteristic and on quantum symplectic manifolds, and is different from the proof given by \textit{T. Bridgeland, A. King} and \textit{M. Reid} [J. Am. Math. Soc. 14, No. 3, 535--554 (2001; Zbl 0966.14028)], where the case of dimension-3 Gorenstein singularities was considered. Bezrukavnikov, R., Kaledin, D.: McKay equivalence for symplectic resolutions of singularities, Tr. Mat. Inst. Steklova \textbf{246} (2004), Algebr. Geom. Metody, Svyazi i Prilozh., 20-42; translation in Proc. Steklov Inst. Math. 2004, no. 3 (246), 13-33 Singularities in algebraic geometry, McKay correspondence, Derived categories of sheaves, dg categories, and related constructions in algebraic geometry McKay equivalence for symplectic resolutions of quotient singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a smooth projective variety and \(D(X)\) the bounded derived category of \(X\). The derived category \(D(X)\) carries a lot of birational information on \(X\). One natural question is to understand the group of isomorphism classes of autoequivalences of \(D(X)\), denoted by \(\text{Auteq\,}D(X)\). Let \(A\) be the subgroup of \(\text{Auteq\,}D(X)\) generated by (1) automorphims of \(X\), (2) tensoring with line bundles and (3) shift functor. When \(K_X\) or \(-K_X\) is ample, Bondal and Orlov showed that \(A\) is the whole autoequivalence group. In general, one expects that \(\text{Auteq\,}D(X)\) contains more elements.
In the paper under review, the authors study the crepant resolution for \(A_n\)-singularities. Let \(Y=\text{Spec} {\mathbb C}[[x,y,z]]/(x^2+y^2+z^{n+1})\) be the \(A_n\)-singularity, \(P \in Y\) the closed point. Consider the crepant resolution \(f: X \to Y\) and \(Z:=f^{-1}(P)\). The authors study the generators of \(\text{Auteq\,}D_Z(X)\), where \(D_Z(X)\) is the ``local derived category'', i.e. the full subcategory of objects supported on the fiber \(Z\). Since \(K_X\) is trivial on the the exceptional divisor \(Z\), one expects there are more elements in \(\text{Auteq\,}D_Z(X)\). Recall that \( \alpha \in D_{Z}(X)\) is spherical if (1) \(\alpha \otimes \omega_X \cong \alpha\), and (2) \( \text{Hom}(\alpha, \alpha)=0\) and \(\text{Hom}^k(\alpha, \alpha)={\mathbb C}\) when \(k \neq 0\). For any spherical object, one can associate a natural twist functor \(T_{\alpha}\). These twist functors play an important role in the study of homological mirror symmetry. Go back to the \(A_n\) case. The exceptional divisor \(Z\) is a chain of rational curves. There are a lot of spherical objects coming from sheaves supported on exceptional curves. Let \(B=\langle T_{\alpha} |\alpha \in D_Z(X), {\text{ spherical}}\rangle\). One is particularly interested in autoequivalences which are given as Fourier-Mukai functors. Consider the subgroup \(\text{Auteq}^{\text{FM}}D_Z(X) \subset \text{Auteq}\,D_Z(X)\) consisting of Fourier-Mukai transforms. (Note that not all autoequivalences of \(D_Z(X)\) are given by Fourier-Mukai transforms.) One of the main results is that the group \(\text{Auteq}^{\text{FM}}D_Z(X)\) is generated by \(B\), Pic\((X)\) and the shift functor. One of the most technical proposition is the following:
Key Proposition. For any \(\Psi \in \text{Auteq\,}D_Z(X)\), there exists an integer \(i\) and \(\Phi \in B\) such that \(\Phi \circ \Psi\) sends every skyscraper sheaf \({\mathcal O}_x\) with \(x \in Z\) to \({\mathcal O}_y[i]\) for some \(y \in Z\).
The main results follow from this proposition and induction. Fourier-Mukai transform; derived category; \(A_n\) singularity; autoequivalence; spherical object Ishii, Akira; Uehara, Hokuto, Autoequivalences of derived categories on the minimal resolutions of \(A_n\)-singularities on surfaces, J. Differential Geom., 71, 3, 385-435, (2005) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Singularities in algebraic geometry, Derived categories, triangulated categories Autoequivalences of derived categories on the minimal resolutions of \(A_n\)-singularities on surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a complex algebraic variety of dimension \(n\) and \(\overline{X}\) be its compactification such that boundary \(\partial X\) has a nonsingular neighborhood in which it is a divisor \(D\) with simple normal crossings. One defines the CW-complex \(\Delta(\partial X)\) whose \(k\)-dimensional simplices correspond to irreducible components of intersections of \(k+1\)-dimensional components of \(D\) and where the inclusions of faces correspond to the inclusions of subvarieties. The complex \(\Delta(\partial X)\) was first introduced and studied by \textit{V. I. Danilov} [Mat. Sb., Nov. Ser. 97(139), 146--158 (1975; Zbl 0321.14010)] that has apparently escaped the attention of the author of the paper under review (as well as of many others working in this area). The English translation of the original paper also has many illuminating notes by its translator F. Zak [Math. USSR, Sb. 26, 137--149 (1976; Zbl 0334.14008)]. One of the main results of Danilov is the homotopy invariance of \(\Delta(\partial X)\).
The paper under review contains several improvements and generalizations of Danilov's results. Thus the author treats the case when the singular locus of \(X\) is not proper -- he considers the simple homotopy type instead of the ordinary homotopy type, and uses a weak factorization theorem to deduce its invariance. Special attention is paid to the case, also considered by Danilov, of the resolution complex associated to a resolution of singularities. For example, the author gives an example of an isolated rational singularity whose resolution is not of the homotopy type of a point (although it is known to be of the rational homotopy type of a point). The author also discusses the relationship (well known under some special restrictive assumptions (see [\textit{V. S. Kulikov} and \textit{P. F. Kurchanov}, in: Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 36, 1--217 (1989; Zbl 0881.14003)]) between the cohomology of the boundary and of the resolution complexes to some parts in the mixed Hodge structure of \(X\). Jensen, A.N.: Gfan, a software system for Gröbner fans and tropical varieties. Available at http://home.imf.au.dk/jensen/software/gfan/gfan.html Minimal model program (Mori theory, extremal rays), Singularities in algebraic geometry, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Boundary complexes and weight filtrations | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 a proof of the second author's ``gluing conjecture'', one of the main steps in his conjectural strategy of proving the categorical geometric Langlands conjecture. The authors work on the spectral side of it which concerns the stack that classifies the local systems for a reductive group. On the automorphic side the gluing conjecture amounts to saying that a certain functor is fully faithful. Most results of the paper only make sense for the ``large'' (unbounded quasi-coherent) derived category.
In terms of technique, the paper develops an apparatus for working with singular support of \(ind\)-coherent sheaves. The notion was introduced in the authors' previous work, but now they work in the category of singularities, where the singular support of an object is a subset of the fiberwise projectivization \(\mathbb{P}\mathrm{Sing}(Y)\). Here \(\mathrm{Sing}(Y)\) is a scheme attached to a local complete intersection scheme \(Y\) and equipped with a \(\mathbb{G}_m\) action. The key result is that the category of singularities of \(Y\) carries the structure of the crystal (roughly, a local system) of categories over \(\mathbb{P}\mathrm{Sing}(Y)\). Given a covering family of quasi-smooth stacks \(Z_i\) over \(Y\) satisfying certain conditions this allows to reduce questions about \(ind\)-coherent sheaves on \(Z_i\) and \(Y\) to more tractable questions about topology of correspondences between \(\mathrm{Sing}(Z_i)\) and \(\mathrm{Sing}(Y)\). This is accomplished by the gluing formalism of the paper, we get a description of the more exotic sheaf on \(Y\) in terms of simpler ones on \(Z_i\), for example, when their singular supports are zero they are quasi-coherent.
More specifically, with the help of the gluing formalism the authors reduce the conjecture to a topological claim about contractibility of certain homotopy types obtained by gluing generalized Springer fibers, which parametrize reductions to various parabolic subgroups of a local system together with a nilpotent infinitesimal symmetry. When the local system is trivial the contractibility follows from the Springer correspondence, in general it follows from the paper's study of the Bruhat-Tits stratification on the generalized Springer fibers. categorical geometric Langlands; singular support of ind-coherent sheaves; category of singularities; crystal of categories; Springer correspondence; Bruhat-Tits stratification Geometric Langlands program (algebro-geometric aspects), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Vector bundles on curves and their moduli The category of singularities as a crystal and global Springer fibers | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Quivers are finite directed graphs (in this paper, at various point, additional assumptions are made). The author opens the paper by telling us that a cluster algebra ``is a subalgebra of the rational function field \(\mathbb Q(x_1,\dots,x_n)\) of \(n\) indeterminates equipped with a distinguished set of variables (cluster variables) grouped into overlapping subsets (clusters) consisting of \(n\) elements, defined by a recursive procedure (mutation) on quivers.'' Also, using the data of a quiver and an associated graded vector space \(W\), the author defines a moduli space \(\mathfrak M(W)\) which he calls a \textit{graded quiver variety}, and a vector space \(\mathbf E_W\) of representations of an associated \textit{decorated quiver}. The first result of this paper states \(\mathfrak M(W)\) is equivalent to \(\mathbf E_W\).
Thereafter, the author defines \(\mathbb L(W)\) and describes it as ``the sum of perverse sheaves [elsewhere denoted \(L(-)\)] whose supports are the whole \(\mathbf E_W^*\).'' Additionally, the author defines what he calls the \textit{truncated \(q\)-character} which he denotes by \(\chi_{q,t}(-)_{\leq 2}\). One of the author's stated main results is an explicit formula for \(\chi_{q,t}(\mathbb L(W))_{\leq 2}\) when \(W\) satisfies certain conditions. Using this formula, the author penultimately gives criterion for \(L(W)\) to corresponds to a cluster monomial (a monomial in the cluster variables from a single cluster). Lastly, with respect to a decomposition \(W=\bigoplus W^i\) (the author calls it the \textit{canonical decomposition}), it is shown that if \(L(W^i)\) are simple and correspond to cluster variables \(w_i\), then \(\bigotimes L(W^i)\) is simple if and only if all \(w_i\) are in a common cluster. quiver variety; cluster algebra H. Nakajima, ''Quiver varieties and cluster algebras,'' Kyoto J. Math., vol. 51, iss. 1, pp. 71-126, 2011. Cluster algebras, Representations of quivers and partially ordered sets, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Quantum groups (quantized enveloping algebras) and related deformations Quiver varieties and cluster algebras | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this paper we study the Grassmannian of submodules of a given dimension inside the radical of a finitely generated projective module \(P\) for a finite dimensional algebra \(\Lambda\) over an algebraically closed field. The orbit of such a submodule \(C\) under the action of \(\operatorname{Aut}_{\Lambda}(P)\) on the Grassmannian encodes information on the top-stable degenerations of \(P / C\). The goal of this article is to begin the study of the global geometry of the closures of such orbits. In dimension one, this geometry is determined by the local rings of singular points. The smallest dimension for which the global geometry is not determined by local data is two, and this case is our main focus. We give several examples to illustrate the interplay between the geometry of the projective surfaces which arise and the corresponding posets of top-stable degenerations. Grassmannian; local rings; projective surfaces Representations of associative Artinian rings, Rational and ruled surfaces, Group actions on varieties or schemes (quotients), Representations of quivers and partially ordered sets Orbit closures and rational surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be a connected semisimple Lie group. There are two natural duality constructions that assign to \(G\): its Langlands dual group \(G^\vee \), and its Poisson-Lie dual group \(G^*\), respectively. The main result of this paper is the following relation between these two objects: the integral cone defined by the cluster structure and the Berenstein-Kazhdan potential on the double Bruhat cell \(G^{\vee ; w_0, e} \subset G^\vee\) is isomorphic to the integral Bohr-Sommerfeld cone defined by the Poisson structure on the partial tropicalization of \(K^* \subset G^*\) (the Poisson-Lie dual of the compact form \(K \subset G\)). By \textit{A. Berenstein} and \textit{D. Kazhdan} [Contemp. Math. 433, 13--88 (2007; Zbl 1154.14035)], the first cone parametrizes the canonical bases of irreducible \(G\)-modules. The corresponding points in the second cone belong to integral symplectic leaves of the partial tropicalization labeled by the highest weight of the representation. As a by-product of our construction, we show that symplectic volumes of generic symplectic leaves in the partial tropicalization of \(K^*\) are equal to symplectic volumes of the corresponding coadjoint orbits in Lie\((K)^*\). To achieve these goals, we make use of (Langlands dual) double cluster varieties defined by \textit{V. V. Fock} and \textit{A. B. Goncharov} [Ann. Sci. Éc. Norm. Supér. (4) 42, No. 6, 865--930 (2009; Zbl 1180.53081)]. These are pairs of cluster varieties whose seed matrices are transpose to each other. There is a naturally defined isomorphism between their tropicalizations. The isomorphism between the cones described above is a particular instance of such an isomorphism associated to the double Bruhat cells \(G^{w_0, e} \subset G\) and \(G^{\vee ; w_0, e} \subset G^\vee \). Langlands duality; Poisson-Lie duality; cluster algebras; potentials; tropicalization General properties and structure of real Lie groups, Cluster algebras, Geometric aspects of tropical varieties, Quantum groups (quantized function algebras) and their representations, Poisson manifolds; Poisson groupoids and algebroids Langlands duality and Poisson-Lie duality via cluster theory and tropicalization | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In the present article, the author pays homage to the work of A.\ L.\ Onishchik by investigating a number of nontrivial extensions of Onishchik's work on group factorizations. Specifically, Onishchik established that if \(G\) is a simple complex algebraic group, \(G_0\) is a real form of \(G\), and \(H\subset G\) is a connected closed complex Lie group such that \(G_0\) acts transitively on \(X=G/H\), then \(H\) is algebraic and \(X\) has one of five possible forms belonging to one of the three following cases: (1) \(X\) is a flag variety, (2) \(X\) is a quasiaffine variety, or (3) \(X\) is affine.
The author considers the extension one obtains by replacing `transitive' by `locally transitive' in Onishchik's result in the case \(X\) is quasiaffine. (The other two cases are known.) This culminates in an exhaustive list of simple complex groups and corresponding real forms that act locally transitively on some quasiaffine variety (theorem 6.2). In order to prove this theorem, the author produces a lower bound \(d_0\) on the codimension of a generic \(G_0\)-orbit in a quasi-affine homogeneous space \(X=G/H\). The bound \(d_0\) is expressed in terms of the rank semigroup of \(X\) (theorem 6.1). In addition to using \(d_0\) to prove theorem 6.2, the author investigates values of \(d_0\) in specific cases. For instance, if \(G\) is a simple group, \(H\subset G\) is a reductive spherical subgroup, and \(X=G/H\), then either \(d_0=\text{ rk}\; X\) or \(d_0=\text{ rk}\; X-1\) (theorem 6.4). reductive algebraic group; quasiaffine variety; spherical subgroup D. N. Akhiezer, Real forms of complex reductive groups acting on quasiaffine varieties, in: Lie Groups and Invariant Theory, Amer. Math. Soc. Transl. Ser. 2, Vol. 213, Providence, RI, 2005, pp. 1-13. Group actions on varieties or schemes (quotients), Semisimple Lie groups and their representations, Complex Lie groups, group actions on complex spaces Real forms of complex reductive groups acting on quasiaffine varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The present paper provides an extension of the theory of perverse sheaves to algebraic stacks and therefore to moduli problems, \(\mathbb{Q}\)-varieties, algebraic spaces, etc. We also include a detailed study of the intersection cohomology of algebraic stacks and their associated moduli spaces. Smooth group scheme actions on singular varieties and the associated derived category turn up as special cases of the more general results on algebraic stacks. -- The main goals of the present paper are as follows:
(i) generalize much of the basic theory of perverse sheaves as done by \textit{A. A. Beilinson}, \textit{J. Bernstein} and \textit{P. Deligne} [``Faisceaux pervers'', Astérisque 100 (1982; Zbl 0536.14011)] to algebraic stacks; as a result the main results on perverse sheaves (for example the decomposition theorems for direct images of perverse sheaves by a proper map) are shown to hold in much more generality and apply in much wider contexts, for example moduli problems, \(\mathbb{Q}\)-varieties, algebraic spaces etc.
(ii) define and study the intersection cohomology of algebraic stacks (and their associated moduli spaces) in arbitrary characteristics. Recall that the only previous study of the intersection cohomology of moduli spaces is by \textit{F. C. Kirwan} [see ``Cohomology of quotients in symplectic and algebraic geometry'', Math. Notes 31 (1984; Zbl 0553.14020) and Invent. Math. 90, 153-167 (1987; Zbl 0631.14012)]; however her study is from an entirely different point of view and is only valid for complex varieties.
(iii) using the observation [see \textit{M. Artin}, Invent. Math. 27, 165-189 (1974; Zbl 0317.14001); p. 180] that algebraic stacks may be viewed as groupoid objects in the category of algebraic spaces, we are able to include the study of smooth group-scheme actions and the associated `equivariant' derived category as a special case of our general study of algebraic stacks. This provides an alternate construction of the equivariant derived category along with all the relevant machinery; the equivariant derived category becomes the natural home of the equivariant intersection cohomology complexes introduced by the author [see \textit{R. Joshua}, Math. Z. 195, 239-253 (1987; Zbl 0637.14014)] and has further applications [see \textit{R. Joshua}, ``Construction of simple (Deligne-Langlands) quotients for representations of \(p\)-adic groups'' (to appear)]. perverse sheaves; algebraic stacks; moduli problems; algebraic spaces; intersection cohomology; group-scheme actions; equivariant derived category Joshua, R.: The intersection cohomology and the derived category of algebraic stacks. In: Algebraic KTheory and Algebraic Topology, NATO ASI Series C, Vol. 407, pp. 91--145, Kluwer Academic, Dordrecht, 1993 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Group actions on varieties or schemes (quotients), Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Derived categories, triangulated categories The intersection cohomology and derived category of algebraic stacks | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(V\) be a finite-dimensional vector space over a field \(K\) of characteristic \(\neq 2\), \(f\) a symmetric or alternating bilinear form on \(V\) and \(O(V)\) or \(\text{Sp} (V)\) the corresponding orthogonal or symplectic group, respectively. For \(\pi \in O(V)\) or \(\text{Sp} (V)\), its path is \(B(\pi)=V(\pi-1)\). One calls \(\pi\) simple if \(\dim B(\pi)=1\). The simple elements in \(O(V)\) are the symmetries (reflections) \(\sigma_ a\) with \(a\) anisotropic; their conjugacy classes are known to correspond to the spinorial norm \(\Theta (\sigma_ a)=f(a,a) K^{*2}\). Analogously for the simple elements in \(\text{Sp} (V)\), which are the symplectic transvections.
It is known that any \(\pi \in O(V)\) can be written as a product of \(k=\dim B(\pi)\) symmetries: \(\pi=\sigma_ 1 \sigma_ 2 \dots \sigma_ k\), provided \(B(\pi)\) is not totally isotropic. It is shown in this paper that, under a mild condition on \(B(\pi)\), the conjugacy classes of \(\sigma_ 2,\dots,\sigma_ k\) can be prescribed in advance; the conjugacy class of \(\sigma_ 1\) is then, of course, determined by the relation
\[
\Theta (\pi)=\Theta (\sigma_ 1) \Theta (\sigma_ 2)\dots \Theta (\sigma_ k).
\]
An analogous result is proved for \(\text{Sp} (V)\), where any \(\pi\) can be written as a product of \(\dim B(\pi)\) symplectic transvections. More detailed results are derived in case the field \(K\) is ordered or Euclidean. orthogonal group; product decomposition; reflections; symplectic group; symmetries; conjugacy class; symplectic transvections Knüppel, F.: Products of simple isometries of given conjugacy types. Forum. math. 5, 441-458 (1993) Orthogonal and unitary groups in metric geometry, Factorization of matrices, Classical groups (algebro-geometric aspects) Products of simple isometries of given conjugacy types | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In Raynaud and Gruson (Invent Math 13:1-89, 1971) and Raynaud (Compos Math 24:11-31, 1972) developed the theory of blowing-up an algebraic variety \(X\) along a coherent sheaf \(M\). However, not much is known about the singularities of the blow-up. In this article, we prove that if \(X\) is a normal surface singularity and \(M\) is a reflexive \({\mathcal{O}}_X\)-module, then such a blow-up arises naturally from the theory of McKay correspondence. We show that the normalization of the blow-up of Raynaud and Gruson is obtained by a resolution of \(X\) such that the full sheaf \(\mathcal{M}\) associated to \(M\) (i.e., the reflexive hull of the pull-back of \(M)\) is globally generated and then contracting all the components of the exceptional divisor not intersecting the first Chern class of \(\mathcal{M}\). Moreover, we prove that if \(X\) is Gorenstein and \(M\) is special in the sense of Wunram (Math Ann 279(4):583-598, 1988) and Riemenschneider (Compos Math 24:11-31, 1972) (generalized in Fernández de Bobadilla and Romano-Velázquez (Reflexive Modules on Normal Gorenstein Stein Surfaces, Their Deformations and Moduli, arXiv:1812.06543, 2018)), then the blow-up of Raynaud and Gruson is normal. Finally, we use the theory of matrix factorization developed by Eisenbud, to give concrete examples of such blow-ups. maximal Cohen-Macaulay module; flatifying blowing-up; Gorenstein surface singularity; matrix factorization Cohen-Macaulay modules, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), McKay correspondence, Complex surface and hypersurface singularities, Local complex singularities On the blow-up of a normal singularity at maximal 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 The article has two aims: (1) to give a tutorial introduction to differential graded Lie algebras, functors of Artin rings and obstructions; (2) To explain ideas and techniques underlying some recent papers concerning vanishing theorems of obstructions to deformations of complex Kähler manifolds.
The author follows the guiding principle proposed by Quillen, Deligne and Drinfeld, that in characteristic \(0\) every deformation problem is governed by a differential graded Lie algebra. In this article the author applies this principle and its methods to prove vanishing theorems for obstruction spaces. The first example is of deformations of a compact complex manifold \(X\) with holomorphic tangent bundle \(\Theta_X\), where the obstruction space can be completely computed using infinitesimal methods, i.e. a deeper study of deformations of \(X\) over fat points, and obstruction theory. It is possible to prove that there exists a well defined vector subspace \(O\subset H^2(X,\Theta_X)\) which is minimal for the property of lifting a particular holomorphic map. The author claims that the paper contains an illustration of how the differential graded Lie algebras can be conveniently used to construct non trivial morphisms of vector spaces \(H^2(X,\Theta_X)\rightarrow W\) such that \(w(O)=0\).
The author starts of by giving the precise definition of differential graded Lie algebras (DGLA's), noticing in particular that the bracket of a DGLA \(L\) induces a structure of graded Lie algebra on its cohomology \(H^\ast(L)=\bigoplus_i H^i(L).\) From the general idea of lifting complexes of vectorspaces \((V,\overline{\partial})\) upto isomorphism, the condition \(\overline{\partial}\circ\overline{\partial}=0\) translates to \(d\xi+\frac12[\xi,\xi]=0,\) and the equivalence of liftings translates to \(\xi^\prime=\xi+\sum_{n=0}^\infty \frac{[a,-]^n}{(n+1)!}([a,\xi]-da).\) So to a DGLA \(L\) the author defines the Maurer-Cartan functor \(MC_L:\mathbf{Art}\rightarrow \mathbf{Set}\) associated to \(L\) given by \(MC_L(A)=\{x\in L^1\otimes\mathfrak m_A|dx+\frac12[x,x]=0\}.\) The equation in the condition is called the Maurer-Cartan equation. The equivalence relation above gives a corresponding action, equivariant on the solution space, called the gauge action. The author then defines the functor \(\text{Def}_L:\mathbf{Art}\rightarrow\mathbf{Set}\) by
\[
\text{Def}_L(A)=\frac{MC_L(A)}{\text{gauge equivalence}}.
\]
Thus in this case, the functor of infinitesimal deformations of a complex \((V,\overline\partial)\) is isomorphic to \(\text{Def}_L\), where \(L\) is the differential graded Lie algebra \(\text{Hom}^\ast(V,V).\) The author recalls the following theorem, sometimes called the basic theorem of deformation theory:
Let \(f:L\rightarrow M\) be a morphism of DGLA's. Then \(f\) induces a natural transformation of functors \(\text{Def}_L\rightarrow\text{Def}_M.\) Moreover, if: \parindent=6mm \begin{itemize}\item[(1)] \(f:H^0(L)\rightarrow H^0(M)\) is surjective; \item[(2)] \(f:H^1(L)\rightarrow H^1(M)\) is bijective; \item[(3)] \(f:H^2(L)\rightarrow H^2(M)\) is injective;
then \(\text{Def}_L\rightarrow\text{Def}_M\) is an isomorphism.
Now, the guiding principle can be stated as follows: Let \(F:\mathbf{Art}\rightarrow\mathbf{Set}\) be the functor of infinitesimal deformation of some algebro-geometric object defined over \(\mathbb{K}\). Then there exists a differential graded Lie algebra \(L\), defined up to quasiisomorphism, such that \(F\simeq\text{Def}_L.\)
The main idea is then to transform a deformation situation with obstruction, to a another deformation situation without obstructions. This is illustrated in the examples of (deformations of) locally free sheaves, complex manifolds, and embedded manifolds.
All in all, the article introduces deformation theory in a good way, and contains a lot of nice and explicit examples. This is a very clever way to do deformation theory, and puts a systematic light on techniques used by a lot of other authors in the field.\end{itemize} differential graded Lie algebras; DGLA; infinitesimal deformation; fundamental theorem of deformation theory; Maurer-Cartan equation; gauge action Marco Manetti, Differential graded Lie algebras and formal deformation theory, Algebraic geometry --- Seattle 2005. Part 2, Proc. Sympos. Pure Math., vol. 80, Amer. Math. Soc., Providence, RI, 2009, pp. 785 -- 810. Noncommutative algebraic geometry Differential graded Lie algebras and formal deformation theory | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(C\) be a singular curve of genus \(g\) over the complex field, with \(g\) nodes as singularities and having a normalization \(\pi:\mathbb{P}^ 1\to C\). It is known [cf. \textit{D. Mumford}, ``Tata lectures on theta. II'', Prog. Math. 43 (1984; Zbl 0549.14014); chapter III b, \S5] that the (generalized) Jacobian \(\text{Jac} C\) is isomorphic to the product of \(g\) copies of the multiplicative group \(\mathbb{C}^*\). This space can be compactified, as analytic space, to \(\text{Jac} C \cong (\mathbb{P}^ 1)^ g/ \sim_ \nu\), where \(\sim_ \nu\) is a suitable equivalence relation determined by a symmetric matrix \(\nu\) such that, for \(1\leq i\), \(j\leq g\), \(\nu_{ii}=0\) and \(\nu_{ij}\) is invertible for \(i\neq j\).
The author proves that, when \(k\) is a ring and \(\nu\) is a symmetric matrix with entries in \(k\), satisfying the above conditions, then the quotient \(A_ \nu\) of \((\mathbb{P}^ 1)^ g \times k\) with respect to the equivalence relation induced by \(\nu\), is a stable quasi-abelian scheme and he explicitly describes that scheme as \(\text{Proj} R\), where \(R\) is a graded ring of totally degenerate theta functions relative to \(\nu\). By means of this description the author proves that the quotient \(A_ \nu\) is a projective scheme and that the canonical map \((\mathbb{P}^ 1)^ g \times k \to A_ \nu\) is finite. generalized Jacobian; totally degenerate theta function; singular curve Azad, H., and J. J. Loeb,Plurisubharmonic functions and Kählerian metrics on complexification of symmetric spaces, Indag. Math. N. S.3(4) (1992), 365--375. Theta functions and abelian varieties On the algebra of totally degenerate theta 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 This paper is devoted to the study of categorical primitive forms of Calabi-Yau categories under the condition that the Hochschild cohomology is semi-simple. In this case the primitive forms are classified by grading operators on the Hochschild homology. For Fukaya categories, the authors showed that closed-open map gives a natural choice and allows the author to recover the genus \(0\) Gromov-Witten invariants, hence enumerative mirror symmetry follows from homological mirror symmetry.
The contents in more detail:
In Section 1 the authors give an introduction to the general framework of producing genus zero enumerative invariants from categories through variation of semi-infinite Hodge structures (VSHS), and states the main results of the paper.
In Section 2 the authors review backgrounds about Hochschild (co)homology of \(A_\infty\) algebras and categories. In particular, they give a proof of the folklore result that Hochschild cohomology of saturated Calabi-Yau category has a natural structure of Frobenius algebra.
In Section 3 the authors prove the first main theorem of the paper: there is a natural bijection between grading operators on Hochschild homology and the set of good splittings of the Hodge filtration compatible with the CY structure.
Section 4 is devoted to the proof of the general bijection between categorical primitive forms of the VSHS associated to a Calabi-Yau category and the good splittings compatible with the CY structure. Associated with the primitive form there is a natural Frobenius structure on the deformation space of the Calabi-Yau category, whose central fiber gives the Frobenius algebra structure on Hochschild cohomology.
In Section 5 the authors apply the previous results to Fukaya categories to extract the Gromov-Witten invariants. The problem is reduced to finding a correct grading on the Hochschild homology which reproduce the Frobenius manifold constructed by Gromov-Witten theory. Assuming standard conjectures in symplectic geometry, the Hochschild homology of Fukaya category is isomorphic to the quantum cohomology, and the pull back of the natural grading (up to re-scaling and shift) satisfies the need. This also allows the authors to derive enumerative mirror symmetry from homological mirror symmetry. categorical primitive forms; mirror symmetry; variation of semi-infinite Hodge structures; categorical enumerative invariants \(A_{\infty}\)-categories, relations with homological mirror symmetry, Variation of Hodge structures (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Categorical primitive forms of Calabi-Yau \(A_\infty\)-categories with semi-simple 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 In the article under review, Poirier generalizes the geometric Langlands correspondence over \(\mathbb C\) from smooth curves to curves with ordinary multiple point singularities.
In the first section, the author works within the category of complex analytic spaces. For an integral projective singular curve \(X\) of positive genus, she considers the image \(\underline{\Omega}\) of the sheaf of holomorphic differentials on \(X\) within the direct image of the sheaf of holomorphic differentials of the normalization of \(X\). She defines a point \(P\) on \(X\) to be a mild singularity of \(X\) if the natural morphism \(\underline{d}_P:\mathcal O_{X,P}\rightarrow\underline{\Omega}_P\) satisfies \(\underline{d}_P\mathfrak m_P^s=\mathfrak m_P^{s-1}\underline{\Omega}_P\) for all \(s\geq 1\), where \(\mathfrak m_P\) denotes the maximal ideal of \(\mathcal O_{X,P}\). If \(P\) is mildly singular, then \(\underline{d}_P\) is surjective. Examples for mild singularities include special singularities, which are singular points \(P\) of \(X\) with the property that the conductor of \(\mathcal O_{X,P}\) equals \(\mathfrak m_P\); ordinary multiple points are special. The author gives an example of a curve \(X_{\text{not mild}}\) with a cusp singularity \(P\) where \(\underline{d}_P\) fails to be surjective. Poirier then defines connections \((\mathcal{M},\nabla)\) on general curves \(X\) as above, using the sheaf \(\underline{\Omega}\) instead of the sheaf of holomorphic differentials \(\Omega^1_{X/\mathbb C}\). For the curve \(X_{\text{not mild}}\), she gives an example of a connection whose kernel is not a local system, thereby showing that the equivalence of categories between connections and local systems, valid for smooth curves, fails for \(X_{\text{not mild}}\). She then proves the first main theorem of her paper, stating that if \(X\) has at most mildly singular points, then the functor \((\mathcal{M},\nabla)\mapsto\ker(\mathcal{M},\nabla)\) establishes an equivalence between the category of connections on \(X\) and the category of local systems on \(X\). Her proof consists in finding, locally at a singular point \(P\) and after a choice of basis for \(\mathcal{M}_P\), a fundamental matrix \(Y\) for \(\nabla_P\). Poirier first constructs a formal solution \(\hat{Y}\) to the problem, with entries in the formal completion \(\hat{\mathcal O}_{X,P}\) of \(\mathcal O_{X,P}\), using the mildness assumption; she then uses general results on differential equations to show that the entries of \(\hat{Y}\) are convergent.
In the second section, Poirier continues to work in the above setting, albeit imposing the stronger assumption that the singularities of \(X\) be at most multiple ordinary points. A local system \(E\) on \(X\) yields, via pullback, a local system \(s^*E\) on the normalization \(s:\tilde{X}\rightarrow X\) of \(X\). To go the opposite way, level structures are required. Poirier defines notions of level structures for local systems and connections on \(\tilde{X}\): a level structure on a local system \(E\) on \(\tilde{X}\) is defined to consist, for each singular point \(P\) of \(X\), of a compatible collection of isomorphisms between the stalks \(E_{P'}\) in the points \(P'\) of the \(s\)-fiber of \(P\). Level structures for connections on \(\tilde{X}\) are defined similarly; here the definition can be extended to any curve with at most special singularities. Poirier observes that the categories of local systems with level structure on \(\tilde{X}\) and connections with level structure on \(\tilde{X}\) are equivalent, and she also establishes an equivalence with the categories of local systems respectively connections on \(X\).
In section 3, Poirier works in the category of algebraic varieties over \(\mathbb C\); as before, she considers integral projective curves of positive genus. She first recalls the fact that given a smooth such curve together with a finite set of positive divisors \(D_1,\dots,D_r\) with disjoint supports, one can construct a curve \(X=Y_{D_1,\dots,D_r}\) with at most special singularities having \(Y\) as a normalization such the support of \(\sum_i D_i\) coincides with the preimage of the singular locus of \(X\). She then defines an equivalence relation on the group of divisors of \(X_{\text{smooth}}\) by declaring that two such divisors be equivalent if and only if they differ by a rational function that is constant modulo the \(D_i\); she defines \(J_{D_1,\dots,D_r}\) to be the resulting group of equivalence classes. Given a divisor \(M\) on \(X_{\text{smooth}}\), the author computes the space of positive divisors on \(X_{\text{smooth}}\) that are equivalent to \(M\) in the above sense, and she gives a criterion for this space being nonempty, using the generalized Riemann-Roch theorem and regular differential forms. Poirier then defines a notion of \((D_1,\dots,D_r)\)-level structure for invertible sheaves on \(\tilde{X}\), and she introduces an equivalence relation on the set of invertible sheaves on \(\tilde{X}\) with level structure. The resulting set \(F(D_1,\dots,D_r)\) of equivalence classes carries a natural group structure, and there is a canonical group homomorphism onto \(\text{Pic}(\tilde{X})\) whose kernel is computed explicitly. Poirier establishes a natural isomorphism \(J_{D_1,\dots,D_r}\overset{\sim}{\rightarrow}F(D_1,\dots,D_r)\); its surjectivity is based on the aforementioned criterion for the existence of certain positive divisors. The author then defines a functor \(\mathcal F(D_1,\dots,D_r)\) from the category of \(\mathbb C\)-schemes to the category of abelian groups whose group of \(\mathbb C\)-valued points is \(F(D_1,\dots,D_r)\); she exhibits a surjection of presheaves from \(\mathcal F(D_1,\dots,D_r)\) onto the relative Picard functor \(\text{Pic}_X\) of \(X\), and she computes its kernel. Using rigidified line bundels with level structure, Poirier shows that \(\mathcal F(D_1,\dots,D_r)\) is representable by a commutative group scheme of finite type \(J_{D_1,\dots,D_r}\) over \(\mathbb C\). The author then shows that \(J_{D_1,\dots,D_r}\) is canonically isomorphic to the Jacobian variety of \(X\); as a corollary, she gives a description of the Jacobian of an arbitrary integral projective curve over \(\mathbb C\) of positive genus, using birational approximation by a special singular curve.
In the fourth and final section, Poirier works in the \(\mathbb C\)-analytic category; she considers integral projective curves over \(\mathbb C\) with positive genus and only ordinary multiple points as singularities, and she generalizes the geometric Hecke correspondence to curves \(X\) of this type. More specifically, she defines, using the explicit description of \(J_{D_1,\dots,D_r}\) given in section 3, a morphism \(H\) from \((\tilde{X}\setminus S)\times J_{D_1,\dots,D_r}\) to \(J_{D_1,\dots,D_r}\), where \(\tilde{X}\) is the normalization of \(X\) and where \(S\subset\tilde{X}\) is the union of the fibers above the singular points of \(X\). She then shows the final main result of the paper, that a rank one connection with level structure on \(\tilde{X}\) admits a `preimage' under \(H\). The proof again uses the explicit structure of \(J_{D_1,\dots,D_r}\) and its relation to the relative Picard functor.
The paper is very well written; Poirier explains the arguments in detail, and her exposition is clear. singular curves; generalised Jacobian; Hecke correspondence Geometric Langlands program (algebro-geometric aspects), Singularities of curves, local rings The geometric correspondence for singular 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 Given an exceptional vector bundle \(E\in\text{coh }\mathbb{X}\), by a theorem of the author and \textit{H. Lenzing} [Categories perpendicular to exceptional bundles (preprint, 1993)], its left-perpendicular category \(^\perp E\) is isomorphic to a category \(\text{mod }\Sigma_E\) for some finite-dimensional hereditary algebra \(\Sigma_E\) having \(n-1\) simple modules (where \(n\) denotes the \(\mathbb{Z}\)-rank of the Grothendieck-group of \(\text{coh }\mathbb{X}\)). In general, the description of \(\Sigma_E\) is a rather difficult and largely unsolved problem. -- For all exceptional vector bundles belonging to an Auslander-Reiten-component of \(\text{coh }\mathbb{X}\) containing a line bundle, the system of indecomposable injective objects in \(^\perp E\) completes \(E\) to a tilting sheaf \({\mathcal I}_E\), whose endomorphism ring \(\text{End}({\mathcal I}_E)\) is a branch-co-extension of a tame concealed quiver \(\Gamma\) of type \(\widetilde\Delta=\widetilde\mathbb{A}_{1,1}\), \(\widetilde\mathbb{D}_4\), \(\widetilde\mathbb{E}_6\), \(\widetilde\mathbb{E}_7\), \(\widetilde\mathbb{E}_8\) by linear quivers (cf. Theorem 3.4).
As a consequence, for each exceptional vector bundle \(E\) belonging to a line-bundle-component, the quiver underlying \(\Sigma_E\) (where \(\text{mod }\Sigma_E\cong{^\perp E}\)) can be determined. -- Theorem 3.3 lists the series of quivers arising in this way. exceptional vector bundles; perpendicular categories; finite-dimensional hereditary algebras; simple modules; Grothendieck groups; Auslander-Reiten components; indecomposable injective objects; tilting sheaves; endomorphism rings; tame concealed quivers Representations of associative Artinian rings, Representations of quivers and partially ordered sets, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers, Representation type (finite, tame, wild, etc.) of associative algebras, Module categories in associative algebras Hereditary module categories arising as categories perpendicular to exceptional 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 \(\Gamma\) be a discrete subgroup of \(\text{SL}_ 2(\mathbb R)\) such that the corresponding modular curve \(X=X(\Gamma)\) has finite volume. It is of interest to study the subgroup \(C(\Gamma)\) of \(J=\text{Jac}(X)\) generated by the divisors of degree \( 0\) supported on the cusps of \(X\). If \(\Gamma\) is a congruence subgroup, then Manin and Drinfeld used the theory of Hecke operators to prove that \(C(\Gamma)\) is finite. A second proof can be given by explicitly constructing modular functions with the appropriate zeros and poles [see \textit{D. S. Kubert} and \textit{S. Lang}, ``Modular units'' (1981; Zbl 0492.12002)].
In this paper the authors give a third proof, based on ideas of \textit{B. Schoeneberg} [``Elliptic modular functions'' (1974; Zbl 0285.10016)], \textit{G. Stevens} [``Arithmetic on modular curves'', Prog. Math. 20 (1982; Zbl 0529.10028)], and \textit{A. J. Scholl} [Math. Proc. Camb. Philos. Soc. 99, 11--17 (1986; Zbl 0564.10023)]. Associated to a cuspidal divisor is a differential of the third kind, which in turn is given by an Eisenstein series of weight 2; and the divisor has finite order in \(J\) if and only if the Eisenstein series has algebraic Fourier coefficients. The authors review this material, and then use Ramanujan sums to obtain an explicit expression for the Fourier coefficients. Since this expression is visibly algebraic, they conclude that \(C(\Gamma)\) is finite.
In case \(\Gamma\) is not a congruence subgroup, it is possible for \(C(\Gamma)\) to be infinite. The authors next consider the well-known realization of the Fermat curve \(F_ N: X^ N+Y^ N=1\) as \(X(\Gamma)\) for a non-congruence subgroup, where the cusps are the \(N\) points ``at infinity''. They find that the finiteness of \(C(\Gamma)\) is equivalent to the algebraicity of a complicated (but very explicit) expression involving generalized Ramanujan sums. Since \textit{D. E. Rohrlich} [Invent. Math. 39, 95--127 (1977; Zbl 0357.14010)] has shown in this case that \(C(\Gamma)\) is finite, the authors conclude that their expression is algebraic.
In the final section the authors look at the (unramified) correspondence between \(F_ N\) and \(X(2N)\) over \(X(2)\) considered by Kubert and Lang (op. cit.). This gives a divisor on the surface \(F_ N\times X(2N)\), and they show that this divisor has finite order in the relative Néron-Severi group \(\text{NS}(F_ N\times X(2N))/(\text{NS}(F_ N)\oplus \text{NS}(X(2N)))\). number of divisors; modular curve; cuspidal divisor; Ramanujan sums; Fourier coefficients; Fermat curve [MR]V.K. Murty andD. Ramakrishnan, The Manin-Drinfeld Theorem and Ramanujan Sums, Proc. Indian Acad. Sci. 97 (1987), 251--262. Arithmetic ground fields for curves, Global ground fields in algebraic geometry, Holomorphic modular forms of integral weight The Manin-Drinfeld theorem and Ramanujan sums | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 purpose of this paper is to prove some results of \textit{Yu. G. Prokhorov} [in: Algebra, Proc. Int. Algebraic Conf., Moscow 1998, 301-317 (2000; Zbl 1003.14005)] for the non-\(\mathbb{Q}\)-factorial case, namely, the existence theorem for the pure log terminal blow-up (theorem 1.5) and the criterion for weakly exceptional singularity (theorem 2.1). These blow-ups allow us to apply the Shokurov inductive method [see \textit{V. V. Shokurov}, J. Math. Sci., New York 102, No. 2, 3876-3932 (2000; Zbl 1177.14078)] to the study of singularities, or more generally, to contractions of algebraic varieties. It reduces the problems of structure, completability, and exceptionality of a singularity to a single exceptional blow-up divisor. For \(\mathbb{Q}\)-factorial singularities, the pure log terminal blow-up is the only blow-up that allows us to extend the complement from the exceptional divisor to the entire variety in the general case (remark 1.3), and for non-\(\mathbb{Q}\)-factorial log terminal singularities, such blow-ups will differ from the pure log terminal blow-up by a small flop contraction (corollary 1.13). In the study of arbitrary \(\mathbb{Q}\)-Gorenstein singularities, it is practically impossible to separate the class of \(\mathbb{Q}\)-factorial singularities from the others. For this reason, one has to use theorems and constructions valid in the general case. The paper under review specifies the results of \textit{Yu. G. Prokhorov} (loc. cit.) about the inductive method for analysis of arbitrary log canonical singularities. log terminal blow-ups; flop С. А. Кудрявцев, ``О чисто логтерминальных раздутиях'', Матем. заметки, 69:6 (2001), 892 -- 898 Global theory and resolution of singularities (algebro-geometric aspects), Minimal model program (Mori theory, extremal rays), Rational and birational maps, Singularities in algebraic geometry Pure log terminal blow-ups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper deals with the global McKay-Ruan correspondence for \(K\)-equivalent orbifolds. An orbifold \(X\) is a complex algebraic variety with quotient singularities. It is called a global quotient if it is of the form \(X=U/G\) for \(U\) smooth and \(G\) a finite group acting on \(U\). An \(n\)-dimensional orbifold is called Gorenstein if all local isotropy groups are finite subgroups of \(\text{SL}_n(\mathbb{C})\). Two Gorenstein orbifolds \(X\) and \(Y\) are called \(K\)-equivalent if there is a common birational resolution \(\phi:Z\rightarrow X\) and \(\psi:Z\rightarrow Y\) such that \(\phi^*K_X=\psi^*K_Y\) (where \(K\) denotes the canonical divisor).
The main result of this paper is a generalization of a theorem of \textit{J. Denef} and \textit{F. Loeser} [Compos. Math. 131, No. 3, 267--290 (2002; Zbl 1080.14001)] for non-global orbifolds.
Theorem. If the orbifolds \(X\) and \(Y\) are \(K\)-equivalent and complete, then their orbifold Hodge numbers, orbifold Hodge structures and orbifold Euler characteristics coincide.
This result appears also in [\textit{T. Yasuda}, Twisted jet, motivic measure and orbifold cohomology, \url{arXiv:math.AG/0110228}]. The proof consists in the computation of the motivic measure \(\mu^{\text{orb}}(\mathcal{L}(X))\).
In Section 2, the authors recall the computation for the case of a global quotient, following \textit{E. Looijenga} [Motivic measures, Astérisque 276, 267--297 (2002; Zbl 0996.14011)]. In Section 3, they generalize the method to the general case by thinking of a general orbifold as a Deligne-Mumford stack given by an étale separated groupoid \(G\) in schemes. They consider the inertia groupoid \(\wedge G\) which extends the usual notion of twisted sectors and perform the decomposition of the space of arcs \(\mathcal{L}(X)\) over the connected components of \(\wedge G\). McKay correspondence; orbifolds; motivic integration; Chen-Ruan conjecture Lupercio, E.; Poddar, M.: The global mckay-ruan correspondence via motivic integration. Bull. London math. Soc. 36, 509-515 (2004) McKay correspondence, Arcs and motivic integration, Generalizations (algebraic spaces, stacks), Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry The global McKay-Ruan correspondence via motivic integration | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Given a Dynkin Diagram \(D\) of finite type, authors construct an affine algebraic variety \(\mathcal{M}_D\), called cluster configuration space of type \(D\), and its partial conmpactification \(\widetilde{\mathcal{M}}_D\), in analogy to what was done in [\textit{F. C. S. Brown}, Ann. Sci. Éc. Norm. Supér. (4) 42, No. 3, 371--489 (2009; Zbl 1216.11079)] for the partial compactification of \(\mathcal{M}_{0,n}\) (recovering this case as a special case when the Dynkin diagram \(D\) is \(A_n\)). They show that \(\mathcal{M}_D\) and \(\widetilde{\mathcal{M}}_D \) are smooth affine algebraic varieties and that the boundary stratification of \(\widetilde{\mathcal{M}}_D\) is simple normal-crossing. They show also that \(\mathcal{M}_D\) can be obtained as a quotient of a cluster variety by the action of the cluster automorphism torus \(T\) considered in [\textit{T. Lam} and \textit{D. E. Speyer}, Algebra Number Theory 16, No. 1, 179--230 (2022; Zbl 1498.13062)], and that \(\widetilde{\mathcal{M}}_D \) can be viewed as an affine open in a projective toric variety.
Exploiting the analogy with \((\mathcal{M}_{0,n})\), the authors show some properties and features about \(\mathcal{M}_D\) and \(\widetilde{\mathcal{M}}_D\).
First, they consider the distinguished nonnegative part \(\mathcal{M}_{D,\geq 0}\) observing that is a stratified space homeomorphic to the face stratification of the generalized associahedron, and then show that the positive part \(\mathcal{M}_{D,>0}\subset \mathcal{M}_D(\mathbb{R})\) is a distinguished connected component which satisfies the other properties of a positive geometry in the sense of \textit{N. Arkani-Hamed} et al. [J. High Energy Phys. 2017, No. 11, Paper No. 39, 124 p. (2017; Zbl 1383.81273)].
Let \(\tilde{B}\) be a full rank acyclic extended exchange matrix of type \(D\), \(\mathcal{A}(\tilde{B})\) its cluster algebra. Then they show that \(\mathcal{M}_D\) is isomorphic to the (free) quotient of the locus where all cluster variables are non-vanishing inside \(\mathrm{Spec}(\mathcal{A}(\tilde{B}))\) by the cluster automorphism group, generalizing the construction of \(\mathcal{M}_{0,n}\) from the Grassmannian \(Gr(2, n)\).
As an application of their results on quotients of cluster varieties and on \(F\)-polynomials, The positive tropicalization of the cluster configuration space \(\mathrm{Trop}_{>0} \mathcal{M}_D\) is identified with the cluster fan \(\mathcal{N}(D^{\vee})\). This answers positive Conjecture 8.1 in the work [\textit{D. Speyer} and \textit{L. Williams}, J. Algebr. Comb. 22, No. 2, 189--210 (2005; Zbl 1094.14048)] on positive tropicalizations of cluster varieties of finite type. Namely from this they show as a corollary that if \(\tilde{B}\) is of full rank and has finite type, then \(\mathrm{Trop}_{>0}X(\tilde{B})/L\) is combinatorially isomorphic to the complete fan \(\mathcal{N}(B)\), \(X(\tilde{B})\) being the cluster variety of \(\mathcal{A}(\tilde{B})\), \(L\) the lineality space of its tropicalization.
They proceed with studying the topology of \(\mathcal{M}_D(\mathbb{C})\) and \(\mathcal{M}_D(\mathbb{R})\), ending in the computation of point counts over finite fields, and the Euler characteristics of \(\mathcal{M}_{B_n}(\mathbb{R})\) and \(\mathcal{M}_{B_n}(\mathbb{C})\)).
In the end they define the cluster string amplitude in analogy with superstring amplitudes on \(\mathcal{M}_{0,n}\), opening to further directions. configuration space; cluster algebras; generalized associahedron; string amplitudes Cluster algebras, Combinatorial aspects of algebraic geometry, Projective and enumerative algebraic geometry, String and superstring theories; other extended objects (e.g., branes) in quantum field theory Cluster configuration spaces of finite type | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Suppose \(C\) is a hyperelliptic curve over a field \(k\) with characteristic avoiding \(2\). For a fixed prime \(\ell\) avoiding the characteristic of \(k\), let \(H^1\) denote the étale cohomology group \(H^1_{\mathrm{ét}}(C_{\bar{k}}, \mathbb{Q}_\ell)\). Given a finite subgroup \(G\) of \(\mathrm{Aut}_k(C)\), it is desirable to explicitly recover the quotient curve \(C/G\) and the associated representation on \(H^1\). In the present article, the authors give a recipe for precisely this computation in the case that an affine model
\[
y^2 = f(x) = c \prod_{r \in R} (x - r)
\]
for \(C\) is known, and that the group \(G\) consists of affine automorphisms -- that is, every \(g \in G\) should act on \(C\) as
\[
g \cdot (x,y) = (\alpha(g) x + \beta, \gamma(g) y).
\]
The results are completely explicit; one may recover an affine model for \(C/G\) basically from a computation of the \(G\)-orbits of \(R\), the roots of \(f(x)\) under \(G\)-action. Similarly, \(H^1 \otimes \mathbb{C}\) can be expressed in terms of the permutation representation \(\mathbb{C}[R]\), the \(G\)-action on points at infinity, and the data \(\{\gamma_g\}_g\). In the case of positive characteristic, the paper also gives an explicit recipe for descending the Frobenius automorphism from \(C\) to \(C/G\) in certain cases. The final section illustrates the explicity use of the theorems with an interesting example involving a hyperelliptic curve of genus \(3\). hyperelliptic curves; quotient curves; étale cohomology Étale and other Grothendieck topologies and (co)homologies, Coverings of curves, fundamental group, Curves over finite and local fields, Curves of arbitrary genus or genus \(\ne 1\) over global fields Quotients of hyperelliptic curves and étale 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 The object of this paper is to determine which combinations T of simple singularities can occur on a quartic surface X in \(P^ 3\). The results are complete if the sum (r) of the Milnor numbers of T is \(\leq 14\); for \(r=15, 16, 17\) they depend on certain restrictions on the root lattice Q of T. Since all components of T are assumed simple, X is a K3 surface, so Q embeds in the Picard group of the resolution \(\tilde X.\) Using the surjectivity of the period map for K3 surfaces, a converse result is established: that if the orthogonal sum S of Q and \({\mathbb{Z}}\lambda\) (where \(\lambda^ 2=-4)\) embeds in the unimodular even lattice \(\Lambda\) with signature (19,3) so that, if \(\tilde S\) is the pure hull of S, then (a) \(\eta\in \tilde S\), \(\eta.\lambda =0\), \(\eta^ 2=2\Rightarrow \eta \in Q,\quad and\) (b) \(u^ 2=0\), \(u.\lambda =-2\Rightarrow u\not\in \tilde S\), then one can construct a surface X. The problem is thus reduced to one in quadratic form theory. The restrictions on r and Q are used to construct isotropic elements in the orthogonal complement \(S^{\perp}\) of S in \(\Lambda\) : \(S^{\perp}\) and the desired embedding can then be reconstructed from local data. The long list of cases that could arise is organised by using the concept of 'elementary transformations' of root systems, which yield subsystems, so that only 9 (maximal) lattices Q need to be embedded: conversely, these are shown to arise using the classification of positive definite irreducible unimodular lattices of rank \(\leq 17\). simple singularities on a quartic surface; period map for K3 surfaces; quadratic form; root systems T. Urabe, ''Elementary transformations of Dynkin graphs and singularities on quartic surfaces,''Invent. Math.,87, 549--572 (1987). Singularities of surfaces or higher-dimensional varieties, Quadratic forms over global rings and fields, Simple, semisimple, reductive (super)algebras, \(K3\) surfaces and Enriques surfaces Elementary transformations of Dynkin graphs and singularities on quartic surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The authors study two sides of non-commutative geometry --- one based on abelian and triangulated categories, the other being replacements of classical schemes. Gabriel proved that any noetherian scheme \(X\) can be reconstructed uniquely up to isomorphisms from the abelian category \(\text{Qcoh} X\) of quasi-coherent sheaves on \(X\). This result has been generalized to quasi-compact schemes by Rosenberg.
Balmer reconstructs a noetherian scheme \(X\) from the triangulated category of perfect complexes \(\mathcal D_{\text{per}}(X)\), and this result has been generalized to quasi-compact, quasi-separated schemes by Buan-Krause-Solberg.
In this article the authors reconstructs affine and projective schemes from appropriate abelian categories. The first result is that for \(R\) (respectively \(A\)) a commutative ring (respectively commutative graded ring, finitely generated by \(A_0\)), the maps
\[
V\mapsto\mathcal S=\{M\in\text{Mod} R~|~\text{supp}_R(M)\subseteq V\},\text{ }\mathcal S\mapsto V=\bigcup_{M\in\mathcal S}\text{supp}_R(M)
\]
(and correspondingly in the graded case) induce bijections between
1) the set of all subsets \(V\subseteq\text{Spec} R\) (respectively \(V\in\text{Proj} A\)) of the form \(V=\bigcup_{i\in\Omega}Y_i\) with \(\text{Spec} R\setminus Y_i\) (respectively \(\text{Proj} A\setminus Y_i)\) quasi-compact and open for all \(i\in\Omega\),
2) the set of all torsion classes of finite type in \(\text{Mod} R\) (respectively tensor torsion classes of finite type in \(\text{QGr} A\)).
This result says that \(\text{Spec} R\) and \(\text{Proj} A\) contain all the information about finite localizations in \(\text{Mod} R\) and \(\text{QGr} A\) respectively. The other result says that there is a 1-1 correspondence between the finite localizations in \(\text{Mod} R\) and the triangulated localizations in \(\mathcal D_{\text{per}}(R)\) :
For a commutative ring \(R\), the map
\[
\mathcal S\mapsto\mathcal T=\{X\in\mathcal D_{\text{per}}(R)|H_n(X)\in\mathcal S\text{ for all } n\in\mathbb Z\}
\]
induces a bijection between
1. the set of all torsion classes of finite type in \(\text{Mod} R\),
2. the set of all thick subcategories of \(\mathcal D_{\text{per}}(R)\).
The final result then, is the reconstruction theorem: \vskip0,2cm \noindent There are natural isomorphisms of ringed spaces
\[
(\text{Spec} R,\mathcal O_R)\overset{\thicksim}\longrightarrow (\text{Spec}(\text{Mod}R),\mathcal O_{\text{Mod} R})
\]
and
\[
(\text{Proj} A,\mathcal O_{\text{Proj} A})\overset{\thicksim}\longrightarrow(\text{Spec}(\text{QGr} A),\mathcal O_{\text{QGr}A}).
\]
The article seems to have been written for an audience not specialists in algebraic geometry. This means that the details of the graded theory and localizations is defined and recalled. This makes the article a very good introduction to the field, and it is more or less self-contained on the algebra level. However, it is less self-contained on the category theory part, but then there are good references so that this makes no problems.
The article contains the definition of torsion classes, the fg-topology, thick subcategories. It may seem a bit unpolished, but is enjoyable reading of important results. torsion class; fg topology; triangulated category; localization; thick subcategory Grigory Garkusha and Mike Prest, Torsion classes of finite type and spectra, \?-theory and noncommutative geometry, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2008, pp. 393 -- 412. Relevant commutative algebra, Noncommutative algebraic geometry Torsion classes of finite type and spectra | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \({\mathfrak O}=G\cdot e\) be the adjoint orbit of a nilpotent element \(e\) in the Lie algebra \(\mathfrak g\) of a complex connected semi-simple Lie group \(G\). The Zariski closure \(\bar {\mathfrak O} \) is an affine algebraic variety. The author proves that the normalisation \(\bar {\mathfrak O}^{\text{norm}}\) of \(\bar{\mathfrak O}\) is Gorenstein and has only rational singularities.
He uses general duality theory to prove the key lemma: Let \(\pi \colon X\to Y\) be proper birational, with \(X\) smooth and \(Y\) normal. Suppose there is a morphism \(\phi\colon {\mathcal O}_ X \to \omega_ X\), inducing an isomorphism \(\pi_ *\phi \colon \pi_ *{\mathcal O}_ X \to \pi_ *\omega_ X\). Then \(Y\) is Gorenstein with rational singularities.
The key lemma is applied to a desingularisation \(X\) with \(G\)-action of \(\bar {\mathfrak O}^{\text{norm}}\); the morphism comes from a \(G\)-invariant section of \(H^ 0(X,\omega_ X)\). The last section gives a direct construction of this section, due to \textit{D. I. Panyushev}, who also proved the main result [see Funct. Anal. Appl. 25, No. 3, 225-226 (1991); translation from Funkts. Anal. Prilozh. 25, No. 3, 76-78 (1991; Zbl 0749.14030)]. Gorensteinness; desingularisation [H1] V. Hinich,Onthe singularities of nilpotent orbits, Israel Journal of Mathematics73 (1991), 297--308. Complete intersections, Global theory and resolution of singularities (algebro-geometric aspects), Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) On the singularities of nilpotent orbits | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Over a field of characteristic 0, the author considers the following class of normal, \(d\)-dimensional singularities \((A,m)\), which he refers to as ``very good'' elliptic singularities: They are purely elliptic (i.e. have plurigenera \(\delta_ m=1\) for \(m\geq 1)\), Cohen Macaulay and of Hodge type \((0,d-1)\) [in the sense of \textit{S. Ishii}, Math. Ann. 270, 541-554 (1985; Zbl 0541.14002) and in Complex analytic singularities, Proc. Semin., Ibaraki/Jap. 1984, Adv. Stud. Pure Math. 8, 165-198 (1987; Zbl 0628.14002)]. For dimension 2, this means just ``simply elliptic''. The author shows: If further \((A,m)\) is a quasihomogeneous hypersurface singularity and \(\dim A\geq 2\), then \(A\) has a canonical model; this is a direct proof, independent of the recent ``minimal model program''.
As an application of this result, the following theorem on the Newton boundary of a quasihomogeneous polynomial is shown:
``Let \(f\in\mathbb{C}[X_ 1,\ldots,X_{d+1}]\) be a quasi-homogeneous polynomial with respect to the weight \(q=(q_ 1,\ldots,q_{d+1})\), where \(0<q_ i\in\mathbb{Q}\) for \(i=1,\ldots,d+1\). Assume that (1) \(\sum^{d+1}_{i=1}q_ i=\deg_ qf=1\), and (2) \(\{f=0\}-\{0\}\) has only rational singularities. -- Then \(\dim_ \mathbb{R}\Gamma(f)=d\), and the point \((1,\ldots,1)\) is contained in the relative interior of \(\Gamma(f)\).''
Further, \(f\) is shown to be the initial form of a ``very good'' elliptic singularity with respect to the canonical filtration. very good elliptic singularities; minimal model; quasihomogeneous hypersurface singularity; canonical model Tomari, M.: The canonical filtration of higher-dimensional purely elliptic singularity of a special type, Invent. math. 104, No. 3, 497-520 (1991) Singularities of surfaces or higher-dimensional varieties, Minimal model program (Mori theory, extremal rays), Hypersurfaces and algebraic geometry, Singularities in algebraic geometry The canonical filtration of higher dimensional purely elliptic singularity of a special type | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The authors use the following notation: \(k\) an algebraically closed field, \((d_1,d_2, \ldots,d_g)\) a vector of positive integers with \(d_{i-1}| d_i\), and \(\text{char} (k)\nmid d_g\). \({\mathcal A}_{(d_1,d_2, \dots,d_g)}\) denotes the coarse moduli space of abelian varieties with polarization \(\phi\) of type \((d_1,d_2, \dots,d_g)\). The main theorem 1.1 of this article is the existence of a canonical isomorphism \({\mathcal A}_{(d_1,d_2, \dots,d_g)} \to {\mathcal A}_{(\frac{d_1d_g}{d_g}, \frac{d_1d_g}{d_{g-1}}, \dots, \frac{d_1d_g}{d_1})}\,.\) When leaving aside the polarization, this isomorphism simply assigns an abelian variety \(A \to S\) its dual variety \(\widehat A \to S\). The main problem is the canonical choice of a polarization on \(\widehat A \to S\). An algebraic approach to the dual polarization of the above type is given in theorem 2.1. If the polarization \(\phi\) is given by a line bundle, i.e. \(\phi = \phi_L\), then the authors show the existence of a line bundle \(\widehat L\) which defines up to some multiple the dual polarization (see theorem 4.1). The line bundle \(\widehat L\) is obtained by the Fourier-Mukai transform from \(L\). dual polarization; moduli spaces of abelian varieties; Fourier-Mukai transform Birkenhake, Christina; Lange, Herbert, An isomorphism between moduli spaces of abelian varieties, Math. Nachr., 253, 3-7, (2003) Algebraic moduli of abelian varieties, classification, Algebraic theory of abelian varieties An isomorphism between moduli spaces of abelian varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In the present paper, the author gives a new interpretation of smooth threefold flops in terms of Fourier-Mukai transforms, thereby relating birational geometry to the theory of perverse coherent sheaves and their moduli.
More precisely, let \(f: Y\to X\) be a birational morphism of projective varieties with 1-dimensional fibres and the property \({\mathbf R}f_*{\mathcal O}_X = {\mathcal O}_X\). Using techniques from the theory of triangulated categories, the author constructs a \(t\)-structure (à la Beilinson-Bernstein-Deligne) on the derived category \(D(Y)\) of coherent sheaves on \(Y\), which then gives rise to the category \(\text{Per}(Y/X)\) of relative perverse sheaves on \(Y\). An object \(F\) of \(\text{Per}(Y/X)\) is called a perverse point sheaf, iff \(F\) is numerically equivalent to the structure sheaf of a point \(y\in Y\), and the author's first main result establishes the fact that there is a projective scheme \({\mathcal M}(Y/X)\) representing the functor of equivalence classes of families of perverse point sheaves in \(\text{Per}(Y/X)\). Next it is shown how these fine moduli spaces \({\mathcal M}(Y/X)\) can be used to derive Fourier-Mukai type equivalences of derived categories. This is then applied to complex projective threefolds with terminal singularities, using the observation that crepant resolutions of terminal threefolds are related by a finite chain of flops, on the one hand, and that flops correspond to Fourier-Mukai transforms at the level of the involved derived categories of coherent sheaves, on the other. In this context, the author's second main theorem concerns birational geometry and reads as follows: If \(X\) is a complex projective threefold with terminal singularities and
\[
\begin{matrix} Y_1 && Y_2\\ f_1\searrow&&\swarrow f_2\\ &X\end{matrix}
\]
are crepant resolutions, then there is an equivalence of derived categories of coherent sheaves \(D(Y_1)@>\sim>> D(Y_2)\).
In particular, this theorem implies that birationally equivalent Calabi-Yau threefolds have equivalent derived categories of coherent sheaves, thereby confirming a conjecture due to \textit{A. I. Bondal} and \textit{D. O. Orlov} [Semiorthogonal decomposition for algebraic varieties, preprint, \texttt{http://arxiv.org/math.A6/9506012}] in full generality.
However, the difficult part in this paper is the construction of the moduli space of perverse point sheaves. This requires several advanced techniques, including geometric invariant theory, stability theory for sheaves, Quot schemes, and the construction of what the author calls the perverse Hilbert scheme \(P\)-Hilb\((Y/X)\). This scheme parametrizes quotients of the structure sheaf of \(Y\) in the category Per\((Y/X)\), and the author's third main theorem establishes its existence as a projective scheme.
All in all, this paper is a major contribution towards a deeper understanding of the significance of Fourier-Mukai transforms and of perverse sheaves within Mori's Minimal Model Program (MMP) in higher-dimensional birational geometry. birational geometry; threefolds; Calabi-Yau threefolds; coherent sheaves; perverse sheaves; moduli spaces; derived categories Bridgeland, Tom. \(Flops and derived categories\). Invent. Math. 147 (2002), no. 3, 613-632. Minimal model program (Mori theory, extremal rays), Derived categories, triangulated categories, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], \(3\)-folds, Calabi-Yau manifolds (algebro-geometric aspects) Flops and derived categories. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities A cusp singularity is a surface singularity whose minimal resolution is a cycle of smooth rational curves meeting transversely. Cusp singularities come in dual pairs. \textit{E. Looijenga} [Ann. Math. (2) 114, 267--322 (1981; Zbl 0509.14035)] proved that if a cusp singularity is smoothable, the minimal resolution of the dual cusp is the anticanonical divisor of some smooth rational surface. The second author and \textit{R. Miranda} [Math. Ann. 263, 185--212 (1983; Zbl 0488.14006)] gave a criterion for smoothability of a cusp singularity, in terms of the existence of a K-trivial semistable model for the central fiber of such a smoothing. We study these ``Type III degenerations'' of rational surfaces with an anticanonical divisor -- their deformations, birational geometry, and monodromy. Looijenga's original paper [loc. cit.] also gave a description of the rational double point configurations to which a cusp singularity deforms, but only in the case where the resolution of the dual cusp has cycle length \(5\) or less. We generalize this classification to an arbitrary cusp singularity, giving an explicit construction of a semistable simultaneous resolution of such an adjacency. The main tools of the proof are (1) formulas for the monodromy of a Type III degeneration, (2) a construction via surgeries on integral-affine surfaces of a degeneration with prescribed monodromy, (3) surjectivity of the period map for Type III central fibers, and (4) a theorem of \textit{N. I. Shepherd-Barron} [Prog. Math. 29, 135--171 (1983; Zbl 0509.14040)] producing a simultaneous contraction to the adjacency. Singularities of surfaces or higher-dimensional varieties, Global theory and resolution of singularities (algebro-geometric aspects) Smoothings and rational double point adjacencies for 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 [For the entire collection see Zbl 0727.00022.]
In the theory of vector bundles on complex algebraic varieties, the so- called exceptional bundles appear as some sort of boundary points in the moduli spaces of stable bundles. Exceptional bundles are defined by cohomological conditions, and the study of them goes back to the work of \textit{J.-M. Drezet} and \textit{J. Le Potier} on stable vector bundles on \({\mathbb{P}}^ 2\) [cf. Ann. Sci. Éc. Norm. Supér., IV. Sér. 18, 193- 243 (1985; Zbl 0586.14007)]. The basic technique for investigating and constructing exceptional bundles has been extended, in the recent years, to arbitrary categories of coherent sheaves on more general algebraic varieties, among others by the author and \textit{A. N. Rudakov} [cf. Duke Math. J. 54, 115-130 (1987; Zbl 0646.14014)]. Nowadays, this technique is well-known as the theory of ``mutations'' or ``helix theory''.
In the present paper, the author develops the theory of mutations from a purely homological (or categorical) point of view. In this general framework based on arbitrary categories of coherent sheaves and their corresponding derived categories, the notions of exceptional objects, mutations of them, mutations of exceptional pairs of them, helices, etc., as well as their basic properties are provided in an algebraically abstract manner, which generalizes (and encompasses) all the technical results obtained in the previous, more specific paper by the author and A. N. Rudakov (cited above). Moreover, this general approach reveals the fundamental essence of the theory of mutations in coherent sheaf theory, makes it possible to overcome the usual problems of existence in concrete particular cases, and allows some more systematic and precise formulations of various results obtained in different contexts. For example, by his method the author is able to recover some results of \textit{M. M. Kapranov} [cf. Invent. Math. 92, No.3, 479-508 (1988; Zbl 0651.18008)] in a different and, apparently, more general way. Another concrete application shows that on some irreducible algebraic surfaces X with \(h^ 0(X,-K_ X)\geq 2\), all the exceptional objects in the derived category of coherent sheaves are given by the twisted images of exceptional sheaves on X, i.e., passing to the level of derived categories does not distort the information content of exceptional sheaves on those surfaces.
As for an even more general account on exceptional pairs, exceptional collections, mutations, and helices in arbitrary categories (instead of specific categories of coherent sheaves on algebraic varieties), with applications to exceptional bundles on algebraic varieties admitting an ample anticanonical class, the reader is referred to \textit{A. N. Rudakov}'s contribution in the same volume [Lond. Math. Soc. Lect. Note Ser. 148, 1-6 (1990; Zbl 0721.14011)]. exceptional bundles; moduli spaces of stable bundles; mutations; helix; derived categories; coherent sheaf theory Gorodentsev A.~L.~Gorodentsev, Exceptional objects and mutations in derived categories, In: Helices and vector bundles: Seminaire Rudakov, London Math.\ Soc.\ Lecture Note Ser., 148, Cambridge Univ.\ Press, Cambridge, 1990, pp.\ 57--73. Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Derived categories, triangulated categories, Nonabelian homological algebra (category-theoretic aspects) Exceptional objects and mutations in derived categories | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let G be a simple linear algebraic group of type \(A_ n\), \(D_ n\) or \(E_ n\) over an algebraically closed field K. We call a unipotent \(x\in G\) p-regular (p\(\geq 0)\) if dim \(Z_ G(x)=2p+rank G\). A 0-regular unipotent is usually called regular and a 1-regular unipotent is called subregular. These two cases have been studied intensively. In this thesis we study 2-regular unipotents.
Let \({\mathcal B}\) denote the variety of Borel subgroups of G, and \({\mathcal B}_ x\) the subvariety of \({\mathcal B}\) consisting of the Borel subgroups containing the fixed unipotent x. For x regular \({\mathcal B}_ x\) consists of one point. Let S be the set of simple roots of G with respect to some fixed Borel subgroup B. For x subregular the irreducible components of \({\mathcal B}_ x\) are lines of type \(\alpha\) (\(\alpha\in S)\), which form together a Dynkin curve of type \(A_ n\), \(D_ n\) or \(E_ n\), as proved by \textit{R. Steinberg} [Conjugacy classes in algebraic groups (Lect. Notes Math. 366, 1974; Zbl 0281.20037), p. 148]. For a 2-regular unipotent x we describe \({\mathcal B}_ x\) in chapters 4 and 5. The irreducible components are either the product of two lines of type \(\alpha\) and \(\beta\) (with \(\alpha\) and \(\beta\) orthogonal simple roots), or they are ruled surfaces with lines of type \(\alpha\) (\(\alpha\in S)\) as fibres. To each irreducible component we can associate a subdiagram of the Dynkin diagram of G and the intersection of two components is determined by the mutual position of the corresponding subdiagrams.
Let \(e(x)=\dim {\mathcal B}_ x\) (hence the unipotent x is e(x)-regular). The Springer representation of the Weyl group W of G on \(H_{2e(x)}({\mathcal B}_ x;{\mathbb{Q}})\) is known to be the trivial representation if x is regular and the standard representation of W as a Coxeter group (tensored with a sign representation) if x is subregular. In chapter 6 we describe the Springer representation and apply it for 2- regular unipotents. simple linear algebraic group; 2-regular unipotents; variety of Borel subgroups; simple roots; irreducible components; Springer representation; Weyl group Linear algebraic groups over arbitrary fields, Representation theory for linear algebraic groups, Classical groups (algebro-geometric aspects), Cohomology theory for linear algebraic groups, Group actions on varieties or schemes (quotients) Borel subgroups containing a fixed 2-regular unipotent. (Thesis) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities An exceptional point in the moduli space \(\mathcal{M}_{g}\) of compact Riemann surfaces of genus \(g\) is a unique surface class whose full automorphism group acts with triangular signature (a group \(G\) acts on a compact Riemann surface \(X\) with triangular signature if the quotient space \(X/G\) has genus \(0\) and the quotient map \(\pi\colon X\rightarrow X/G\) is branched over exactly three points). A surface \(X\) is said to be symmetric if it admits an anticonformal involution, and it is said to be \(k\)-hyperelliptic if it admits a conformal involution, called the \(k\)-hyperelliptic involution, with quotient a curve of genus \(k\). In the special case where \(k=1\), the quotient curve is an elliptic curve and we call \(X\) an elliptic-hyperelliptic curve. Let \(\mathcal{M}^{k}_{g}\) denote the \(k\)-hyperelliptic locus (the surface classes of \(\mathcal{M}_{g}\) admitting a \(k\)-hyperelliptic involution). In the paper under review, the authors prove a number of interesting results about exceptional points, especially those which are also symmetric, in \(\mathcal{M}_{g}^{1}\).
First, the authors consider the problem of determining each group \(G\) up to topological equivalence which can act with a triangular signature on a symmetric surface \(X\) and which contains a \(1\)-hyperelliptic involution for genus \(g> 5\). The purpose for the restriction on the genus is that it guarantees the subgroup generated by the \(1\)-hyperelliptic involution is normal in \(G\) and so simplifies the calculations. To determine such groups, the authors utilize some previous results on the classification of topological equivalence classes of elliptic-hyperelliptic actions [\textit{E. Tyszkowska}, J. Algebra 288, No. 2, 345--363 (2005; Zbl 1078.30037)], together with computational methods from \textit{D. Singerman} [Math. Ann. 210, 17--32 (1974; Zbl 0272.30022)] which provides complete conditions for when \(G\) acts on a symmetric surface.
Next the authors show that for infinitely many different \(g\), the number of exceptional points in \(\mathcal{M}_{g}^{1}\) is larger than any preassigned integer \(n\), though for any \(g\), at most four are also symmetric (the authors also show that for infinitely many \(g\), the number of exceptional points is \(0\)). This is in stark contrast to the hyperelliptic locus, \(\mathcal{M}_{g}^{0}\) where the number of exceptional points is always between \(3\) and \(5\), and is precisely \(3\) for all \(g>30\), all of which are symmetric [see \textit{A. Weaver}, Geom. Dedicata 103, 69--87 (2004; Zbl 1047.32012)]. It is also in contrast to the general \(k\)-hyperelliptic locus \(\mathcal{M}_{g}^{k}\) where for sufficiently high \(g\), there are no exceptional points. The proof of this result arises from the construction of arbitrarily many different isomorphism classes of such \(G\)-actions for infinitely many \(g\), and then refers to the previous results to illustrate that at most four are symmetric.
The authors finish by determining the full group of conformal and anticonformal automorphisms of symmetric exceptional points in \(\mathcal{M}_{g}^{1}\). If \(X\) is a symmetric surface whose surface class is an exceptional point in \(\mathcal{M}_{g}^{1}\), then the existence of a symmetry on \(X\) implies that the full group of conformal and anticonformal automorphisms is a cyclic \(2\) extension of the full group of conformal automorphisms of \(X\). The possible full groups of comformal automorphisms of such a surface were determined in [\textit{E. Tyszkowska}, loc. cit.] (though with an error which has been corrected in the paper under review). The authors use these facts and explicit computations to determine the full groups of conformal and anticomformal automorphisms. elliptic-hyperelliptic surfaces; symmetries of a surface E. Tyszkowska, A. Weaver, \textit{Exceptional points in the elliptic-hyperelliptic locus}, J. Pure Appl. Algebra 212 (2008), 1415--1426. Automorphisms of curves, Compact Riemann surfaces and uniformization, Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) Exceptional points in the elliptic-hyperelliptic locus | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 works on a family of FJRW potentials which is viewed as the counterpart of a nonconvex Gromov-Witten potential via the Landau-Ginzburg/Calabi-Yau correspondence. The author provides an explicit formula for Polishchuk and Vaintrob's virtual cycle in genus 0, constructed algebraically by using matrix factorizations [\textit{A. Polishchuk} and \textit{A. Vaintrob}, J. Reine Angew. Math. 714, 1--122 (2016; Zbl 1357.14024)]. In the nonconcave case of the so-called chain invertible polynomials \(W=x_1^{a_1}x_2+\cdots+x_{N-1}^{a_{N-1}}x_N+x_N^{a_N+1}\), it yields a compatibility theorem between Polishchuk and Vaintrob's class and FJRW virtual class, and also a proof of mirror symmetry for FJRW theory when the chain polynomial is of Calabi-Yau type, where the B-side concerns the local system given by the primitive cohomology of the fibration \([\{W^\vee-t\prod x_j=0\}/\underline{SL}(W^\vee)]_t\to \Delta ^*\) over a punctured disk. Here \(W^\vee=y_1^{a_1}+y_1y_2^{a_2}+\cdots+y_{N-1}y_N^{a_N+1}\) and \(\underline{SL}(W^\vee)\) is a group containing automorphisms of \(W^\vee\) of determinant 1. mirror symmetry; FJRW theory; nonconcavity; virtual cycle; matrix factorization; LG model; spin curves; recursive complex; chain invertible polynomial; Givental's formalism; J-function Guéré, Jérémy: A Landau-Ginzburg mirror theorem without concavity. (2013) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Families, moduli of curves (algebraic), Mirror symmetry (algebro-geometric aspects) A Landauc-Ginzburg mirror theorem without concavity | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 summarizes recent developments, related to the global presentation and strong normal presentation of Picard bundles on Jacobians of curves, especially the results of \textit{G. Pareschi} and \textit{M. Popa}, [see e.g. J. Am. Math. Soc. 13, No. 3, 651--664 (2000; Zbl 0956.14035), Math. Ann. 340, No. 1, 209--222 (2008; Zbl 1131.14049)]. For an abelian variety \(X\), denote by \(\hat{X} = \text{Pic}^{o}(X)\) its dual, and let \(R\Phi: D(X) \rightarrow D(\hat{X})\) be the Fourier-Mukai functor. A coherent sheaf \(F\) on \(X\) is \(IT_i\) (resp. \(WIT_j\)) if \(h^i(F\otimes\alpha)=0\) (resp. \(R^i\Phi(F) = 0\)) for \(i \not=j\) and any \(\alpha \in \text{Pic}^{o}(X)\). For a \(WIT_j\) sheaf \(F\) the complex \(\hat{F} := R\Phi(F)\) is in fact a sheaf (called a Fourier-Mukai transform of \(X\)). If \(X = J = J(C)\) is a Jacobian of a curve \(C\), a Picard bundle is the Fourier-Mukai transform of a line bundle on a special subvariety \(W_d = \{ L \in \text{Pic}^{d}(C): h^0(L) > 0 \}\), \(1 \leq d \leq g = g(C)\) of \(J\). The most of the paper is devoted to a systematic presentation of the results of Pareschi and Popa (ibid.) about the Picard bundles \(E^{n,k} =\widehat{{\mathcal O}_J(n\Theta_J)} \otimes {\mathcal O}_{\hat{J}}(k\Theta_{\hat{J}})\) and \(E_{W_d}^{n,k} =\widehat{{\mathcal O}_{W_d}(n\Theta_J)} \otimes {\mathcal O}_{\hat{J}}(k\Theta_{\hat{J}})\).
By using the new tools developed by Pareschi and Popa, in Section 4 the authors reprove a result of \textit{G. R. Kempf} [Proc. Am. Math. Soc. 108, No. 1, 59--67 (1990; Zbl 0704.14022)]:
The Picard bundles \(E^{n,k}\) (\(n \geq 2\)) and \(E_{W_d}^{2,k}\) are globally generated for any \(k \geq 2\).
In Sections 5 and 6, further properties of the Picard bundles \(E^{2,k}\) and \(E^{2,k}_{W_d}\) related to their strong normal presentation with respect to the theta bundles on the Jacobian \(J\) are studied. Picard bundles; Jacobians Jacobians, Prym varieties, Vector bundles on surfaces and higher-dimensional varieties, and their moduli Effective results on Picard bundles via \(M\)-regularity | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(V\) be a regular neighborhood of a negative chain of \(2\)-spheres (ie an exceptional divisor of a cyclic quotient singularity), and let \(B_{p,q}\) be a rational homology ball which is smoothly embedded in \(V\). Assume that the embedding is simple, ie the corresponding rational blowup can be obtained by just a sequence of ordinary blowups from \(V\). Then we show that this simple embedding comes from the semistable minimal model program (MMP) for \(3\)-dimensional complex algebraic varieties under certain mild conditions. That is, one can find all simply embedded \(B_{p,q}\)'s in \(V\) via a finite sequence of antiflips applied to a trivial family over a disk. As applications, simple embeddings are impossible for chains of \(2\)-spheres with self-intersections equal to \(-2\). We also show that there are (infinitely many) pairs of disjoint \(B_{p,q}\)'s smoothly embedded in regular neighborhoods of (almost all) negative chains of \(2\)-spheres. Along the way, we describe how MMP gives (infinitely many) pairs of disjoint rational homology balls \(B_{p,q}\) embedded in blown-up rational homology balls \(B_{n,a}\sharp \overline{\mathbb{CP}^2}\) (via certain divisorial contractions), and in the Milnor fibers of certain cyclic quotient surface singularities. This generalizes results of Khodorovskiy (2012, 2014), H Park, J Park and D Shin (2016) and Owens (2018) by means of a uniform point of view. antiflip; Mori sequence; rational homology ball Embeddings in differential topology, Differentiable structures in differential topology, Deformations of singularities, Minimal model program (Mori theory, extremal rays), General topology of 4-manifolds Simple embeddings of rational homology balls and antiflips | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\subset\text{SL}(3,\mathbb{C})\) be a finite group. A \(G\)-cluster is a \(G\)-invariant subscheme \(Z\subset \mathbb{C}^3\) of dimension zero with global sections \(H^0(\mathcal{O}_Z)\) isomorphic as a \(\mathbb{C}[G]\)-module to the regular representation \(R\) of \(G\). \textit{I. Nakamura} [J. Algebr. Geom. 10, No.4, 757--779 (2001; Zbl 1104.14003)] introduced the moduli space \(G\text{-Hilb}\) of \(G\)-clusters on \(\mathbb{C}^3\) as a natural candidate for a projective crepant resolution of \(\mathbb{C}^3/G\) and proved it for \(G\) abelian. \textit{T. Bridgeland, A. King} and \textit{M. Reid} [J. Am. Math. Soc. 14, No. 3, 535--554 (2001; Zbl 0966.14028)] subsequently proved the conjecture for all \(G\) by establishing an equivalence of derived categories: \(D(G\text{-Hilb})\sim D^G(\mathbb{C}^3)\).
This paper generalises the notion of \(G\)-cluster: a \(G\)-constellation is a \(G\)-equivariant coherent sheaf \(F\) on \(\mathbb{C}^3\) with global sections \(H^0(F)\) isomorphic as a \(\mathbb{C}[G]\)-module to the regular representation \(R\) of \(G\). Set:
\[
\Theta:=\left\{\theta\in\text{Hom}(R(G),\mathbb{Q})\mid \theta(R)=0\right\}.
\]
For \(\theta\in \Theta\), a \(G\)-constellation is said to be stable (resp. semistable) if every proper \(G\)-equivariant coherent subsheaf \(0\subset E\subset F\) satisfies \(\theta(E)>0=\theta(F)\) (resp. \(\geq\)). Generalizing ideas of \textit{A. V. Sardo-Infirri} [Resolutions of orbifold singularities and the transportation problem on the McKay quiver, preprint, \url{arXiv:alg-geom/9610005}] and \textit{A. D. King} [Q. J. Math., Oxf. II. Ser. 45, 515--530 (1994; Zbl 0837.16005)], the authors study the moduli spaces \(\mathcal{M}_{\theta}\) (resp. \(\overline{\mathcal{M}}_{\theta})\) of \(\theta\)-stable (resp. semistable) constellations. Note that \(G\text{-Hilb}\cong \mathcal{M}_{\theta}\) for parameter \(\theta\) in the cone \(\Theta_+:= \{\theta\in\Theta\mid \theta(\rho)>0\) if \(\rho\neq \rho_0\}\), where \(\rho_0\) denotes the trivial representation of \(G\). A parameter \(\theta\in \Theta\) is generic if every \(\theta\)-semistable \(G\)-constellation is \(\theta\)-stable.
The method of Bridgeland-King-Reid generalises to show that: If \(\theta\) is generic, there is an equivalence of categories \(D(\mathcal{M}_{\theta})\sim D^G(\mathbb{C}^3)\) and \(\mathcal{M}_{\theta}\rightarrow \mathbb{C}^3/G\) is a projective crepant resolution of singularities. It is then natural to ask whether every projective crepant resolution may be realised as a moduli space \(\mathcal{M}_{\theta}\) for some parameter \(\theta\). The main result of this paper answers this question affirmatively in the abelian case: For a finite abelian subgroup \(G\subset \text{SL}(3,\mathbb{C})\), suppose that \(Y\rightarrow \mathbb{C}^3/G\) is a projective crepant resolution. Then \(Y\cong \mathcal{M}_{\theta}\) for some parameter \(\theta\).
For generic \(\theta\), put \(C:=\{\eta\in \Theta\mid\) every \(\theta\)-stable \(G\)-constellation is \(\eta\)-stable\}. This is a convex polyhedral cone (or chamber) in \(\Theta\). The subset \(\Theta^{\text{gen}}\subset \Theta\) of generic parameter is open, dense and is the disjoint union of finitely many open convex polyhedral cones in \(\Theta\). For generic \(\theta\), the moduli space \(\mathcal{M}_{\theta}\) depends only upon the open chamber \(C\subset \Theta\) containing \(\theta\), so we write \(\mathcal{M}_C\) in place of \(\mathcal{M}_{\theta}\) for any \(\theta\in C\). Then the proof's idea is as follows: Since every projective crepant resolution is obtained by a finite sequence of flops from \(G\text{-Hilb}\), it is enough to show that, if \(Y\cong \mathcal{M}_C\) for some chamber \(C\), then for any flop \(Y'\) of \(Y\) there is a chamber \(C'\) (not necessarily adjacent to \(C\)) such that \(\mathcal{M}_{C'}\cong Y'\). Then the first step is to understand the walls of chambers in \(\Theta\) (\S 3) and then how the moduli \(\mathcal{M}_C\) changes as \(\theta\) passes through a wall from \(C\) to another chamber \(C'\). The method uses the description of chambers in terms of Fourier-Mukai transforms. Hilbert schemes of orbits; constellations; crepant resolution; Fourier-Mukai; toric geometry Craw-Ishii A.~Craw and A.~Ishii, Flops of \(G\)-Hilb and equivalences of derived categories by variation of GIT quotient, Duke Math.\ J., 124 (2004), 259--307. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Derived categories, triangulated categories, Geometric invariant theory Flops of \(G\)-Hilb and equivalences of derived categories by variation of GIT quotient | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(V\) be a simple finite-dimensional \(G\)-module where \(G\) is a reductive group. All structures are defined over the complex numbers \(\mathbb{C}\). We consider the following problem: What is \(\Aut_G(V)\), the group of polynomial automorphism \(V@>\sim>>V\) that commute with the \(G\)-action?
Consider the non-simple \(SL_2\)-module \(R_4\oplus R_2\), where \(R_k\) denotes the binary forms of degree \(k\). The map \((p,q)\mapsto(p+q^2,q)\) is an \(SL_2\)-equivariant nonlinear automorphism. However, for a simple module it is not easy to give such an automorphism.
We show that the natural representation of \(SL_3\times SL_5\times SL_{13}\) allows nonlinear equivariant automorphisms; more exactly, the group of polynomial automorphisms on \(\mathbb{C}^3\otimes\mathbb{C}^5\otimes\mathbb{C}^{13}\) commuting with the simple \(SL_3 \times SL_5\times SL_{13}\)-action is isomorphic to \(\mathbb{C}^*\ltimes\mathbb{C}\). This is the first example of a simple module with nonlinear equivariant automorphisms.
An appendix is added which shows a relation between the equivariant automorphism group and a rationality question of the linearization problem in terms of a non-abelian cohomology. polynomial automorphism; equivariant automorphism Group actions on varieties or schemes (quotients), Complex Lie groups, group actions on complex spaces, Birational automorphisms, Cremona group and generalizations Nonlinear equivariant automorphisms. -- Appendix: Equivariant automorphisms and the linearization problem | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We consider discrete subgroups \(\Gamma\) of the simply connected Lie group \({\widetilde {\text{SU}}}(1,1)\) of finite level. This Lie group has the structure of a 3-dimensional Lorentz manifold coming from the Killing form. \(\Gamma\) acts on \({\widetilde {\text{SU}}}(1,1)\) by left translations. We want to describe the Lorentz space form \(\Gamma\setminus{\widetilde {\text{SU}}}(1,1)\) by constructing a fundamental domain \(F\) for \(\Gamma\). We want \(F\) to be a polyhedron with totally geodesic faces. We construct such \(F\) for all \(\Gamma\) satisfying the following condition: The image \(\overline\Gamma\) of \(\Gamma\) in \(\text{PSU}(1,1)\) has a fixed point \(u\) in the unit disk of order larger than the level of \(\Gamma\). The construction depends on \(\Gamma\) and \(\Gamma u\). For co-compact \(\Gamma\) the Lorentz space form \(\Gamma\setminus{\widetilde{ \text{SU}}}(1,1)\) is the link of a quasi-homogeneous Gorenstein singularity. The quasi-homogeneous singularities of Arnold's series \(E\), \(Z\), \(Q\) are of this type. We compute the fundamental domains for the corresponding group. They are represented by polyhedra in Lorentz 3-space shown in Tables 1--13. Lorentz space form; polyhedral fundamental domain; quasihomogeneous singularity; Arnold singularity series Brieskorn, E.; Pratoussevitch, A.; Rothenhäusler, F., The combinatorial geometry of singularities and arnold's series E, Z, Q, Mosc. Math. J., 3, 273-333, (2003) Complex surface and hypersurface singularities, Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics, Singularities of surfaces or higher-dimensional varieties, Fuchsian groups and their generalizations (group-theoretic aspects), Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization), Teichmüller theory for Riemann surfaces The combinatorial geometry of singularities and Arnold's series \(E\), \(Z\), \(Q\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 toric singularities of the form of \(\mathbb{C}^4/\Gamma\) for finite abelian groups \(\Gamma\subset \text{SU}(4)\). In particular, we consider the simplest case \(\Gamma=\mathbb{Z}_2 \times\mathbb{Z}_2 \times \mathbb{Z}_2\) and find explicitly charge matrices for partial resolutions of this orbifold by extending the method by Morrison and Plesser. We obtain three kinds of algebraic equations, \(z_1 z_2 z_3 z_4=z_5^2, z_1 z_2 z_3=z_4^2 z_5\) and \(z_1 z_2 z_5 = z_3 z_4\) where \(z_i\)'s parametrize \(\mathbb{C}^5\). When we put \(N\) D1 branes at this singularity, it is known that the field theory on the worldvolume of \(N\) D1 branes is \(T\)-dual to \(2 \times 2 \times 2\) brane cube model. We analyze geometric interpretation for field theory parameters and moduli space. Ahn, C-h; Kim, H., Branes at \( {\text{\mathbb{C}}}^4/\Lambda \) singularity from toric geometry, JHEP, 04, 012, (1999) Quantum field theory; related classical field theories, Special varieties Branes at \({\mathbb{C}}^4/\Gamma\) singularity from toric geometry | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The Bogomolov-Tian-Todorov theorem states that a non-singular \(n\)-fold \(X\) with \(c_1(X)=0\) has unobstructed deformation theory, i.e. the moduli space of \(X\) is non-singular. This theorem was reproven using algebraic methods by \textit{Z. Ran} [J. Algebr. Geom. 1, No. 2, 279-291 (1992; Zbl 0818.14003)]. Since then, it has been proven for Calabi-Yau \(n\)-folds with various mild forms of isolated singularities: ordinary double points by Kawamata and Tian, Kleinian singularities by Ran, and finally, in the case of threefolds, arbitrary terminal singularities by Namikawa. Now, the most natural class of singularities in the context of Calabi-Yau \(n\)-folds are canonical singularities. Indeed, if \(X\) is a Calabi-Yau \(n\)-fold with terminal singularities, and \(f:X\to Y\) is a birational contraction, \(Y\) normal, then \(Y\) has canonical singularities. Thus the natural question to ask is: Is the deformation space of Calabi-Yau \(n\)-folds with canonical singularities unobstructed?
It appears worthwhile to give a counterexample to this most general question. We give an example of a Calabi-Yau \(n\)-fold \(X\) with the simplest sort of dimension 1 canonical singularities, and show that \(X\) lies in the intersection of two distinct families of Calabi-Yau \(n\)-folds. One is a family of generically non-singular Calabi-Yau's, and the other is a family of Calabi-Yau's which generically have terminal singularities. (In the case \(n=3\), these are also nonsingular.) In particular, the point of the moduli space corresponding to \(X\) is in the intersection of two components of moduli space, and hence has obstructed deformation theory. obstruction of deformation; Calabi-Yau \(n\)-fold; canonical singularities Mark Gross, The deformation space of Calabi-Yau \?-folds with canonical singularities can be obstructed, Mirror symmetry, II, AMS/IP Stud. Adv. Math., vol. 1, Amer. Math. Soc., Providence, RI, 1997, pp. 401 -- 411. Calabi-Yau manifolds (algebro-geometric aspects), Deformations of singularities, Local deformation theory, Artin approximation, etc., Singularities of surfaces or higher-dimensional varieties The deformation space of Calabi-Yau \(n\)-folds with canonical singularities can be obstructed | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 an introduction to several problems arising in singularity theory that can be computed using the Computer Algebra system SINGULAR. The reader can learn how to compute in SINGULAR a wide collection of useful tools to the study of singularities, such as the singular locus, normalization and primary decomposition. For isolated singularities it is possible to compute for example the monodromy, spectral numbers, and Milnor and Tjurina numbers; and for families of singularities the author shows how to construct versal families and deformations. The last part of the paper states the main steps to make an implementation of Villamayor algorithm of resolution of singularities, as an example of a complex theoretical algorithm that can be implemented for SINGULAR. This implementation is already included as a standard library of SINGULAR (release 3.0). It uses identification of points that appear in more than one affine chart of the resolution tree in order to simplify the output of Villamayor algorithm. All these tools are made clear with computable examples that can be checked in practice. The author also includes the necessary theoretical background and a complete list of references where to find more details about the different concepts involved. Computation of invariants in Singularity theory; Resolution of singularities Frühbis-Krüger, A., Computational aspects of singularities, (), 253-327 Global theory and resolution of singularities (algebro-geometric aspects), Computational aspects in algebraic geometry, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) Computational aspects 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 Given a finite quiver \(Q\) and two dimension vectors \(\mathbf{v}\) and \(\mathbf{w}\) one can construct a quiver variety \(\mathfrak{M}\left( \mathbf{v,w}\right) .\) To any any admissible automorphism \(a\) of \(Q\) one can also construct a diagram automorphism \(\theta\) of \(\mathfrak{M}\left( \mathbf{v,w}\right) \) and consider the subvariety of fixed points \(\mathfrak{M}\left( \mathbf{v,w}\right) ^{\theta}.\) The main result in this work is a description of this subvariety, showing that, in the case where the quiver is type \(A\), the subvariety \(\mathfrak{M}\left( \mathbf{v,w}\right) ^{\theta}\) is a disjoint union of quiver varieties for quivers of type \(D\). Moreover, if we take a quiver of type \(A_{2n-1}\) and pick \(\mathbf{v}\) and \(\mathbf{w}\) to be symmetric under the Dynkin diagram involution, then \(\mathfrak{M}\left( \mathbf{v,w}\right) ^{\theta}\) is isomorphic to a resolution of a closed subvariety of a certain Slodowy slice. This allows for a description of the Springer resolution of the corresponding Slodowy slices as a quiver variety of type \(D.\) quiver varieties; automorphisms Henderson, Anthony; Licata, Anthony, Diagram automorphisms of quiver varieties, Adv. Math., 267, 225-276, (2014) Coadjoint orbits; nilpotent varieties, Representation theory for linear algebraic groups, Group actions on varieties or schemes (quotients) Diagram automorphisms of quiver varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a smooth complex projective variety, \(D\) a closed subset of \(X\) and \(\mathbf{H}\) a strict normal crossing divisor on \(X\). Let \(U=X - D\). Given a triangulation \(K\) of \(X\) we can define the set of \(\delta-\)admissible chains \(AC_p(K,D,\mathbb{Z})\) as in Definition 2.5 of this paper. By taking a direct limit on the triangulations over the corresponding complex \(AC_{\bullet}(K,D,\mathbb{Z})\), we get \(AC_{\bullet}(X,D,\mathbb{Z})\). It turns out that this complex is quasi isomorphic to the relative complex defined by simplexes on \(K\). As a result, one of the main theorems of this paper is that there is an isomorphism
\[
H^{j}(U,\mathbf{H},\mathbb{Z})\simeq H^{j}(AC^{\star}(U,\mathbf{H},\mathbb{Z}))
\]
for any non-negative integer \(j\). The complex \(AC^{\star}(U,\mathbf{H},\mathbb{Z})\) is a simple complex associated to a double complex constructed using \(\delta\)-admissible chains on the components of the normal crossing divisor. Moreover, by using integrals over \(\delta-\)admissible chains, the second main theorem of this paper is the existence of a pairing
\[
H^{2\dim U - j}(U,\mathbf{H},\mathbb{Q}) \otimes H^{j}(X - \mathbf{H},D,\mathbb{C})\to \mathbb{C}.
\]
To be more precise, the pairing comes from a pairing between \(\delta\)-admissible chains and smooth forms on \(U\) with compact support and logarithmic singularity along \(\mathbf{H}\).
The author applies the above results to describe the Abel-Jacobi map for higher Chow groups, using the cubical version. For \(Y\) a smooth complex projective variety, the map
\[
\Psi^{p,n}:z_{\mathrm{hom}}^{p}(Y,n)\to J^{p,n}(Y)
\]
admits a description considering admissible chains on \(Y\times (\mathbb{P}^1)^n\). Using a divisor given by the faces of the cubes, there are spaces \(\square^n\) and \(\partial \square^n\) such that the Jacobian \(J^{p,n}(Y)\) can be described in terms of the cohomology of the pair \((\square^n\times Y, \partial \square^n\times Y)\). This is crucial to define the Abel-Jacobi map explicitly using currents and admissible chains. Finally, an example is worked out for polylogarithms as a way to show the Abel-Jacobi map for open varieties. higher Chow groups; abel-jacobi map; admissible chains Semialgebraic sets and related spaces, (Equivariant) Chow groups and rings; motives, Transcendental methods, Hodge theory (algebro-geometric aspects), Algebraic cycles Semi-algebraic chains on projective varieties and the Abel-Jacobi map for higher Chow cycles | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 category of Brauer diagrams, analogous to Turaev's tangle category, is introduced, a presentation of the category is given, and full tensor functors are constructed from this category to the category of tensor representations of the orthogonal group \(\mathrm{O}(V)\) or the symplectic group \(\mathrm{Sp}(V)\) over any field of characteristic zero. The first and second fundamental theorems of invariant theory for these classical groups are generalised to the category theoretic setting. The major outcome is that the authors obtain presentations for the endomorphism algebras of the module \(V^{\otimes r}\) which are new in the classical symplectic case and in the orthogonal and symplectic quantum case, while in the orthogonal classical case, the proof is more natural than in their earlier work. These presentations are obtained by appending to the standard presentation of the Brauer algebra of degree \(r\) one additional relation. This relation stipulates the vanishing of a single element of the Brauer algebra which is quasi-idempotent, and which we describe explicitly both in terms of diagrams and algebraically. In the symplectic case, if dim \( V = 2n\), the element is precisely the central idempotent in the Brauer subalgebra of degree \(n +1\), which corresponds to its trivial representation. Since this is the Brauer algebra of highest degree which is semisimple, the generator is an exact analogue for the Brauer algebra of the Jones idempotent of the Temperley-Lieb algebra. In the orthogonal case the additional relation is also a quasi-idempotent in the integral Brauer algebra. Both integral and quantum analogues of these results are given, the latter of which involve the BMW algebras. Brauer category; invariant theory; second fundamental theorem; quantum group Lehrer, G. and Zhang, R., ' The Brauer category and invariant theory', \textit{J. Eur. Math. Soc.}17 ( 2015) 2311- 2351. Classical groups (algebro-geometric aspects), Vector and tensor algebra, theory of invariants The Brauer category 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 This paper gives a self-contained, detailed account of the construction and compactification of the moduli space of Higgs bundle on (families of) curves. It is divided into five parts: I. The theorem of the cube; II. \(G\)-bundles; III. Abelianisation; IV. Projective connections; V. Infinitesimal parabolic structure.
Part I starts with a theorem on the determinant of the cohomology of coherent sheaves on a curve. It is shown that the theorem of the cube follows from this result. A second application is at the basis of the construction, via theta-functions, of global sections of determinant bundles on the moduli space of Higgs bundles. More precisely, let \(S\) be a noetherian base scheme for the family of curves \(\pi:C\to S\), with \(\pi\) proper, all fibers of dimension \(\leq 1\), and such that \(\pi_ *({\mathcal O}_ C)={\mathcal O}_ S\). A Higgs bundle on \(C\) is a vector bundle \({\mathcal F}\) together with a section \(\theta\) of \(\Gamma(C,{\mathcal E}nd({\mathcal F})\otimes\omega_ C)\). The coefficients of the characteristic polynomial of \(\theta\) define global sections \(f_ i\in\Gamma(C,\omega^ i)\), and the affine space classifying such sections is called the characteristic variety \({\mathcal C}har\) (it depends on \(\text{rk}({\mathcal F}))\), and \({\mathcal F}\) defines a point \(\text{char}({\mathcal F})\) in \({\mathcal C}har\). Such a Higgs bundle will often be denoted \(({\mathcal F},\theta)\). One has the notion of (semi)-stability for Higgs bundles, and any semistable Higgs bundle admits a Jordan- Hölder (JH) filtration by subbundles with stable quotieents of constant ratio degree/rank. The isomorphism classes and multiplicities of these stable components are independent of the filtration. Two semi-stable Higgs bundles are called JH-equivalent if these coincide. A result on JH- equivalence is derived and used to show that the theta-functions separate points in the moduli space of Higgs bundles. The moduli-space of stable Higgs bundles of given rank and degree is constructed as an algebraic space \({\mathcal M}^ 0_ \theta\). Then \({\mathcal M}_ \theta^ 0\) embeds as an open subscheme into the onrmalization \({\mathcal M}_ \theta\) of \(\mathbb{P}^ N\times{\mathcal C}har\) (for suitable \(N)\) in \({\mathcal M}^ 0_ \theta\).
In part II one considers a reductive connected algebraic group \({\mathcal G}\) over a smooth projective connected curve \(C\) over a field \(k\). A \({\mathcal G}\)-torsor \(P\) on \(C\), together with an element
\[
\theta\in\Gamma(C,\text{Lie}({\mathcal G}_ P)\times\omega_ C)
\]
is called semistable if \((\text{Lie}({\mathcal G}_ P)\), ad\((\theta))\) is a semistable Higgs bundle of degree zero. One also has the notion of stable \(P\). The main result on semistable pairs \((P,\theta)\) is the following semistable reduction theorem: If \(V\) is a complete discrete valuation ring with fraction field \(K\), \(C\to V\) a smooth projective curve, \((P_ K,\theta_ K)\) a semistable pair (associated with a connected reductive group \({\mathcal G}\) over \(C)\) whose characteristic is integral over \(V\), then there exists a finite extension \(V'\) of \(V\) such that the base extension of \((P_ K,\theta_ K)\) extends to a semistable pair on \(C_{V'}\). Furthermore, if the special fiber of this extension is stable, then any other semistable extension is isomorphic to it. For stable \((P,\theta)\) one is led to construct an algebraic moduli stack \({\mathcal M}^ 0_ \theta({\mathcal G})\) and the coarse moduli space \(M^ 0_ \theta({\mathcal G})\) which is shown to be quasi-projective of explicitly calculated relative dimension over a suitable base. As before one defines a \(M_ \theta({\mathcal G})\) as the normalisation of a \(\mathbb{P}^ N\) in \(M^ 0_ \theta({\mathcal G})\). Then \(M_ \theta({\mathcal G})\) is projective over \({\mathcal C}har\) and contains \(M^ 0_ \theta({\mathcal G})\) as an open subscheme. Then, for example, if \(C\) has genus \(>2\), the boundary \(M_ \theta({\mathcal G})-M^ 0_ \theta({\mathcal G})\) has codimension \(\geq 4\). Many other results are derived.
In part III the theory is extended to exceptional groups. As a corollary of the theory one obtains, with the notations above, that the set of connected components of the moduli space \({\mathcal M}^ 0({\mathcal G})\) of stable (Higgs) \({\mathcal G}\)-bundles coincides with that of \({\mathcal M}^ 0_ \theta({\mathcal G})\), \(M_ \theta({\mathcal G})\) as well as that of a generic fiber of \({\mathcal M}^ 0_ \theta({\mathcal G})\to{\mathcal C}har\), under the natural mappings. Among many other results, one application of abelianisation is given by another corollary: On each connected component of \({\mathcal M}^ 0_ \theta({\mathcal G})\), all global functions are obtained by pullback from \({\mathcal C}har\).
In part IV the accent is on \({\mathcal M}^ 0({\mathcal G})\), where \({\mathcal G}\) is the twisted form of some semi-simple \(G\). The notion of \(\Omega_ C\)- connections \(\nabla\) on \({\mathcal G}\)-torsors \(P\) is introduced. \({\mathcal M}^ 0_ \nabla({\mathcal G})\) denotes the moduli stack of such pairs \((P,\nabla)\) with \(P\) stable. It is fibered over \({\mathcal M}^ 0({\mathcal G})\). Over \(\mathbb{C}\), \({\mathcal M}^ 0_ \nabla({\mathcal G})\) classifies bundles with integrable connections, i.e. representations of \(\pi_ 1(C)\). A locally faithful \({\mathcal G}\)-representation \({\mathcal F}\) defines a line bundle \({\mathcal L}={\mathcal L}({\mathcal F})\) on \({\mathcal M}^ 0({\mathcal G})\). Then the pullback of \({\mathcal L}\) to \({\mathcal M}^ 0_ \nabla({\mathcal G})\) has a connection \(\nabla\). Its curvature can be described explicitly.
The final part V discusses parabolic structures in the sense of C. Seshadri. The parabolic analogue of a Higgs bundle is introduced and a theory parallel to the one in the foregoing parts is sketched. torsor; moduli space of Higgs bundle; determinant of the cohomology of coherent sheaves on a curve; theorem of the cube; characteristic variety; theta-functions; semistable pairs; moduli stack; abelianisation; connection Faltings, Gerd, Stable {\(G\)}-bundles and projective connections, Journal of Algebraic Geometry, 2, 3, 507-568, (1993) Families, moduli of curves (algebraic), Vector bundles on curves and their moduli, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) Stable \(G\)-bundles and projective connections | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 the modular group, i.e., \(\Gamma =\mathrm{PSL}_{2}(\mathbb{Z})\). Let \(\mathrm{rep}_{n}\Gamma\) be the \(n\)-dimensional representations of \(\Gamma\), and let \(\mathrm{iss}_{n}\Gamma\) be the affine GIT\ quotient \(\mathrm{iss} _{n}\Gamma =\mathrm{rep}_{n}G/PGL_{n}\). Then \(\mathrm{iss}_{n}\Gamma\) describes the isomorphism classes of semisimple \(\Gamma\)-representations of dimension \(n\), and \(\mathrm{iss}_{n}\Gamma\) decomposes into a (disjoint) union of irreducible components \(\mathrm{iss}_{\alpha}\Gamma\), where \(\alpha =(a,b;x,y,z)\), \(a+b=n=x+y+z\) is a dimension vector. With this notation, if \(xyz\neq 0\) and \(\max(x,y,z)\leq \min(a,b)\) then \(\mathrm{iss}_{\alpha}\Gamma\) contains an open subset of simple representations and has dimension \(1+n^{2}-(a^{2}+b^{2}+x^{2}+y^{2}+z^{2})\), a quantity denoted \(d_{\alpha}\).
The principal result of this paper is the construction, for each component \(\mathrm{iss}_{\alpha}\Gamma\) of \(\mathrm{iss}_{n}\Gamma\) which contains irreducible representations, of an é tale rational map \(\mathbb{A}^{d_{\alpha}}\dashrightarrow \mathrm{iss}_{\alpha}\Gamma\) whose image contains a Zariski open dense subset of \(\mathrm{iss}_{\alpha}\Gamma\). The map is made explicit, however the definition is different for the cases \(a=b\) and \(a\neq b\). This result extends works of \textit{I. Tuba} and \textit{H. Wenzl} [Pac. J. Math. 197, No. 2, 491--510 (2001; Zbl 1056.20025)], who proved it for \(n\leq 5\), and \textit{L. Le Bruyn} [J. Pure Appl. Algebra 215, No. 5, 1003--1014 (2011; Zbl 1260.20058)] for \(n\leq 11\). This provides insight as well into the the irreducible representations of the \(3\)-string braid group \(B_{3}\) as their classification reduces to the classification of irreducible \(\Gamma\)-representations. modular group; representations of the modular group; quiver representations; linear dynamical systems Geometric invariant theory, Representations of quivers and partially ordered sets, Trace rings and invariant theory (associative rings and algebras) Bulk irreducibles of the modular group | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper under review is a sequel of [\textit{M. Domokos, H. Lenzing}, J. Algebra 228, No. 2, 738-762 (2000; Zbl 0955.16015)] where the authors investigated the relative invariants and moduli spaces introduced by \textit{A. D. King} [Q. J. Math., Oxf. II. Ser. 45, No. 180, 515-530 (1994; Zbl 0837.16005)]. Let \(k\) be an algebraically closed field, let \(\Sigma\) be a finite-dimensional \(k\)-algebra, and let \(\text{mod }\Sigma\) be the category of finite-dimensional right \(\Sigma\)-modules. The authors study in detail the module category when \(\Sigma\) is a concealed-canonical algebra. In fact the class of such algebras consists of representation-infinite algebras of tame or wild representation type. They prove that for any admissible weight all the corresponding moduli spaces are isomorphic to a certain projective space. As a consequence the authors show that in the case of a tame concealed algebra any infinite moduli space for families of modules is a projective space and all fields of rational invariants on irreducible components of representation spaces are purely transcendental. In particular, a generalization of a result of \textit{C. M. Ringel} [Invent. Math. 58, 217-239 (1980; Zbl 0433.15009)] for the rational invariants of extended Dynkin quivers is obtained. finite-dimensional algebras; representations; concealed-canonical algebras; tame concealed algebras; relative invariants; admissible weights; semistability; coarse moduli spaces; extended Dynkin quivers; perpendicularity DOI: 10.1006/jabr.2001.9117 Representations of quivers and partially ordered sets, Representation type (finite, tame, wild, etc.) of associative algebras, Module categories in associative algebras, Group actions on varieties or schemes (quotients) Moduli spaces for representations of concealed-canonical 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 0575.00008.]
Author's preface: The title ''tangency and duality over arbitrary fields'' was given by \textit{A. H. Wallace} to the article in Proc. Lond. Math. Soc., III. Ser. 6, 321-342 (1956; Zbl 0072.160) in which he pioneered the study of the similarities and differences that appear in the indicated theory, a basic topic in projective algebraic geometry, when the characteristic \(p\) of the ground field is allowed to become positive. The title is also an apt choice for the present work. However, the words ''over arbitrary fields'' were dropped for two reasons: First, the subject has matured to the point where it can, fairly, go without saying that p will be arbitrary. Second and more important, the similarities and differences attendant to p are secondary to the geometry itself. The bulk of material is, moreover, characteristic free, and many of the special considerations required when \(p>0\) highlight features of geometry over the complex numbers that are sometimes taken for granted. A case in point is provided by the central notion of reflexivity. In most situations, instead of assuming \(p=0\), it suffices to assume that the principal varieties are reflexive. A major issue then, when \(p>0\), is to find useful conditions guaranteeing reflexivity.
The present work is an expanded version of the minicourse of 3 lectures given by the author. The work is intended first and foremost to introduce this lovely subject and, in particular, to announce and to introduce a fair number of recent results. An attempt has been made to place the results in context and in perspective, to explain their meaning and significance, and to give a feeling for their proofs. The full details of the proofs, especially if they are available elsewhere, are seldom presented. The presentation is usually expository, rarely formal. There are, however, several mathematical tidbits that are not found elsewhere. Some of these are: a fuller discussion of Wallace's construction of infinitely many plane curves with a given dual curve; a simpler and more conceptual proof of the theorem of generic order of contact, I-(10); a more traditional proof of the reviewer's theorem comparing the ranks of a variety with those of a general hyperplane section and those of a general projection; an account of Landman's unpublished application of Lefschetz theory of the theory of the dual variety; the application of Goldstein's theory of the second fundamental form to the study of the simplicity of a general contact between two varieties one varying; a new derivation of the number, 51, of conics tangent, when \(p=2\), to 5 general conics; and a new study of the limiting behavior of the tangent hyperplanes to a variety degenerating under a homolography. tangency; duality; characteristic p; dual curve; homolography Kleiman, S. , Tangency and duality , in: '' Proc. 1984 Vancouver Conference in Algebraic Geometry '', CMS-AMS Conference Proceedings , pp. 163-226, Vol. 6, 1985. Enumerative problems (combinatorial problems) in algebraic geometry, Projective techniques in algebraic geometry Tangency and duality | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Objects of consideration are pairs (D\&X), where D is a reduced effective divisor on a complete variety X. Two pairs (D\&X), (C\&Y) are said to be (birationally) equivalent if there exist a birational map \(f:\quad X\to Y\) and a one-to-one correspondence \(D_ i\leftrightarrow C_ i\) between irreducible components of D and C such that \(f| D_ i:\quad D_ i\to C_ i\) is also a birational map for each i. In each equivalence class of (D\&X) there exists a non-singular model (Z\&V). Using this model, the author defines generalized plurigenera \(P_ m(\sum a_ iD_ i\&X)\) and the Kodaira-Iitaka dimension \(\kappa\) (\(\sum a_ iD_ i\&X)\) (where \(a_ i\in {\mathbb{Q}}\), \(0\leq a_ i\leq 1)\) and proves independence of these numbers of a choice of the nonsingular model. The rest of the paper is devoted to the pairs of the form (C\&\({\mathbb{P}}^ 2)\), where C is an irreducible plane curve. The author solves the following problem: when is the pair (C\&\({\mathbb{P}}^ 2)\) equivalent to the pair of the form (a non-singular curve \& \({\mathbb{P}}^ 2)?\) He proves that such an equivalence is valid if and only if either \(\kappa(C\&{\mathbb{P}}^ 2)=-\infty\) or the number \(d=(3+\sqrt{8g(C)+1})/2\) is integer, \(d\geq 3\), \(\kappa((3/d)C\&{\mathbb{P}}^ 2)=P_ d((2/d)C\&{\mathbb{P}}^ 2)=P_ d((3/d)C\&{\mathbb{P}}^ 2)-1=0.\) It is worth mentioning the following corollary: \((C\&{\mathbb{P}}^ 2)\sim(a line \& {\mathbb{P}}^ 2)\) iff \(\kappa(C\&{\mathbb{P}}^ 2)=-\infty.\) Another application is a sufficient condition on multiplicities of singular points of C in the presence of which C cannot be transformed into a non-singular plane curve by any birational transformation of \({\mathbb{P}}^ 2\). birational pairs; generalized plurigenus; effective divisor; Kodaira- Iitaka dimension Suzuki, Birational geometry of birational pairs 32 (1983) Rational and birational maps, Divisors, linear systems, invertible sheaves Birational geometry of birational pairs | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(V\) be a \(4\)-dimensional complex vector space, and fix a \textit{complex reflection group} \(W<\mathrm{GL}_\mathbb{C}(V)\), i.e. a finite subgroup of \(\mathrm{GL}_\mathbb{C}(V)\) generated by reflections of order \(2\). The goal of the paper is, for a homogeneous \(W\)-invariant polynomial \(f\), to study the quotient of the zero set \(\mathcal{Z}(f)\subseteq \mathbb{P}(V)\) by some special subgroup \(\Gamma < W\). In particular the authors consider \(\Gamma\) to be either the subgroup \(W^{\mathrm{SL}}=\mathrm{Ker}(\mathrm{det}) \cap W\) or the derived subgroup \(W'\) of \(W\).
The main result of the paper states that, for specific choices of the degree \(d\) of \(f\), and if \(\mathcal{Z}(f)\) has only ADE singularities, then the quotient \(\mathcal{Z}(f)/\Gamma\) is a \(K3\) surface with only ADE singularities.
This allows the authors to classify all the \(K3\) surfaces that can be obtained as a quotient by the subgroups \(W'\) and \(W^{\mathrm{SL}}\). The result builds on previous work by \textit{W. Barth} and \textit{A. Sarti} [Asian J. Math. 7, No. 4, 519--538 (2003; Zbl 1063.14047)], which only focused on a smaller class of such examples of \(K3\) surfaces.
The proof involves a detailed study of the singularities of such quotients, based partly on the theory of Springer and Lehrer-Springer on complex reflection groups. \(K3\) surfaces; complex reflection groups \(K3\) surfaces and Enriques surfaces, Families, moduli, classification: algebraic theory, Reflection and Coxeter groups (group-theoretic aspects) Complex reflection groups and \(K3\) surfaces. I | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Given an affine toric variety \(X_\sigma\) provided by a rational, polyhedral cone \(\sigma \subseteq \mathbb{R}^n\), it is well known that equivariant resolutions of the singularities of \(X_\sigma\) may be obtained by suitable polyhedral subdivisions of \(\sigma\). The new rays plugged into the cone correspond to the exceptional divisors. The author asks for essential divisors for \(X_\sigma\), i.e. those exceptional divisors occurring in every (even non-equivariant) resolution. Since exactly the internal non-splittable elements of \(\sigma \cap \mathbb{Z}^n\) join every polyhedral subdivision into a smooth fan, they should be considered good candidates for being at least equivariantly essential. In fact, as the author proves in the present paper, they provide exactly the essential divisors even in the general sense. Finally, she generalizes her result to divisors mapping not onto Sing \(X_\sigma\), but onto some given closed, equivariant subset of \(X_\sigma\). affine toric variety; equivariant resolution of the singularities; essential divisors; exceptional divisors C. Bouvier, Germes de courbes tracées sur une variété torique singulière et diviseurs essentiels , C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), no. 4, 387-390. Toric varieties, Newton polyhedra, Okounkov bodies, Divisors, linear systems, invertible sheaves, Global theory and resolution of singularities (algebro-geometric aspects) Germs of curves on a singular toric variety and essential divisors. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(R\) be a Noetherian ring, let \(E^\bullet\) be a complex of finitely generated \(R\)-modules, bounded above, such that \(H^i(E^\bullet)\) are finitely generated \(R\)-modules for all \(i\) as well. Then there exists a complex \(F^\bullet\) of finitely generated free \(R\)-modules with the differential \(d\), and a morphism of complexes \(F^\bullet \rightarrow E^\bullet\) which is a quasi-isomorphism. By definition, the \(i\)-th cohomology jump ideal of \(E^\bullet\) is defined to be the ideal of minors of size \(\mathrm{rank}(F^i)-k+1\) of a matrix, representing the direct sum \(d^{i-1}\oplus d^i\) of two components of the differential \(d\). The authors study the affine subscheme \(\mathrm{Spec}(R)\) associated with such an ideal. In fact, this setup occurs in various settings, involving the theory of differential graded Lie algebra pairs. The presented approach is mainly based on standard techniques of general deformation theory developed by \textit{M. Schlessinger} [Trans. Am. Math. Soc. 130, 208--222 (1968; Zbl 0167.49503)] and his successors. The authors also consider a few applications to local systems, vector bundles, Higgs bundles, representations of fundamental groups, and discuss relations with results obtained earlier by many authors (see, e.g. [\textit{W. M. Goldman} and \textit{J. J. Millson}, Publ. Math., Inst. Hautes Étud. Sci. 67, 43--96 (1988; Zbl 0678.53059)], [\textit{A. Dimca} and \textit{Ş. Papadima}, Commun. Contemp. Math. 16, No. 4, Article ID 1350025, 47 p. (2014; Zbl 1315.14006)]). deformation theory; differential graded Lie algebras; cohomology jump loci; local systems; vector bundles; Higgs bundles; representations of fundamental groups Budur, N. and Wang, B., ' Cohomology jump loci of differential graded Lie algebras', \textit{Compos. Math.}151 ( 2015) 1499- 1528. MR3383165. Formal methods and deformations in algebraic geometry, Deformations of fiber bundles, Homotopy theory and fundamental groups in algebraic geometry Cohomology jump loci of differential graded 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 The author gives a reader-friendly introduction to the McKay correspondence, based on his earlier article [Hokkaido Math. J. 32, No. 2, 317--333 (2003; Zbl 1046.14002)]. Most proofs are omitted, but several examples and explicit calculations are included, thus guiding the reader along the historical development through the various levels of this beautiful subject: From the original construction of McKay (including comprehensive calculations in the tetrahedral, octahedral and icosahedral case) to more geometric approaches, culminating in the results of \textit{Y. Ito} and \textit{I. Nakamura} [Proc. Japan Acad., Ser. A 72, No. 7, 135--138 (1996; Zbl 0881.14002)] and the interpretation in terms of derived categories [\textit{A. Ishii}, J. Reine Angew. Math. 549, 221--233 (2002; Zbl 1057.14057); \textit{T. Bridgeland, A. King} and \textit{M. Reid}, J. Am. Math. Soc. 14, No. 3, 535--554 (2001; Zbl 0966.14028)]. Group actions on varieties or schemes (quotients), Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties, Parametrization (Chow and Hilbert schemes), Derived categories, triangulated categories McKay correspondence for quotient 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 In the 1970's, Springer established a parametrization of the irreducible representations of the Weyl group in terms of unipotent conjugacy classes in reductive groups over an algebraically closed field. Since then, the Springer correspondence has been generalized in a number of ways. For instance, Lusztig's generalization of this correspondence depended on the construction of certain intersection cohomology complexes and led to the development of a theory of character sheaves.
\noindent Let \(k\) be an algebraic closure of a finite field of odd characteristic. Let \(V\) be a \(k\)-vector space of dimension \(2n\). Let \(H\) be the symplectic group \(G^{\theta} \cong\mathrm{Sp}_{2n}\) where \(\theta\) is an involutive automorphism on \(G = \mathrm{GL}(V)\). If \(G_{\mathrm{uni}}^{-} := \{g \in G : g~ \text{unipotent},~ \theta(g)=g^{-1} \},\) then \(G_{\mathrm{uni}}^{-} \times V\) is known as Kato's exotic nilpotent cone. Kato showed that \(H\) acts on the exotic nilpotent cone with finitely many orbits and, the orbits are parametrized by the set \(P_n\) of double partitions of \(n\). The Springer correspondence between the irreducible representations of the Weyl group of type \(C_n\) and the \(H\)-orbits of the exotic nilpotent cone was established by Kato using Ginzburg's theory of affine Hecke algebras.
\noindent In this paper, the authors develop a theory of character sheaves for the exotic symmetric space \(G/H \times V\) and give an alternate proof of Kato's result. More precisely, the authors prove the following version of the correspondence. Let \(B\) be a \(\theta\)-stable Borel subgroup of \(G\) and \(U\), its unipotent radical. Let \(M_1, \dots, M_n\) be a flag in \(V\) whose \(H\)-stabilizer is \(B^{\theta}\). If
\[
G_{\mathrm{uni}}^{-} := \{g \in G : g~ \text{unipotent},~ \theta(g)=g^{-1} \},
\]
then consider the variety
\[
\chi_{\mathrm{uni}} := \{(x,v,hB^{\theta}) \in G_{\mathrm{uni}}^{-} \times V \times H/B^{\theta} : h^{-1}xh \in U^{-}, h^{-1}(v) \in M_n \}.
\]
Write \(\pi\) for the projection of \(\chi_{\mathrm{uni}}\) on the first two factors \(G_{\mathrm{uni}}^{-} \times V\), the exotic nilpotent cone. For each double partition \(\lambda\) of \(n\), let \(V_{\lambda}\) denote the irreducible representation of the Weyl group and let \(O_{\lambda}\) denote the corresponding \(H\)-orbit in the exotic nilpotent cone \(\mathcal{N}\). The Weyl group acts on the semisimple perverse sheaf \(\pi_{!} \overline{\mathbb{Q}_l}[\mathcal{N}]\); the authors prove the decomposition:
\[
\pi_{!} \overline{\mathbb{Q}_l}[\mathcal{N}] \cong \oplus_{\lambda \in P_n} V(\lambda) \otimes IC(\bar{O_{\lambda}},\bar{\mathbb{Q}_l})[\dim O_{\lambda}].
\]
Springer correspondece; Kato's exotic nilpotent cone; intersection cohomology; exotic character sheaves T. Shoji, K. Sorlin, \textit{Exotic symmetric space over a finite field,} I, Transform. Groups \textbf{18} (2013), 877-929. Classical groups (algebro-geometric aspects), Representation theory for linear algebraic groups, Linear algebraic groups over finite fields Exotic symmetric space over a finite field. I | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Consider a compact symplectic sub-orbifold groupoid \(\mathsf{S}\) of a compact symplectic orbifold groupoid \((\mathsf{X},\omega)\). Let \(\underline{\mathsf{X}}_{\mathfrak{a}}\) be the weight-a blowup of \(\mathsf{X}\) along \(\mathsf{S}\), and \(\mathsf{D}_{\mathfrak{a}}=\mathsf{PN}_{\mathfrak{a}}\) be the exceptional divisor, where \(\mathsf{N}\) is the normal bundle of \(\mathsf{S}\) in \(\mathsf{X}\). In this paper we show that the absolute orbifold Gromov-Witten theory of \(\underline{\mathsf{X}}_{\mathfrak{a}}\) can be effectively and uniquely reconstructed from the absolute orbifold Gromov-Witten theories of \(\mathsf{X}\), \(\mathsf{S}\) and \(\mathsf{D}_{\mathfrak{a}}\), the natural restriction homomorphism \(H^*_{\mathrm{CR}}(\mathsf{X})\rightarrow H^*_{\mathrm{CR}}(\mathsf{S})\) and the first Chern class of the tautological line bundle over \(\mathsf{D}_{\mathfrak{a}}\). To achieve this we first prove similar results for the relative orbifold Gromov-Witten theories of \((\underline{\mathsf{X}}_{\mathfrak{a}}\mid\mathsf{D}_{\mathfrak{a}})\) and \((\underline{\mathsf{N}}_{\mathfrak{a}}\mid\mathsf{D}_{\mathfrak{a}})\). As applications of these results, we prove an orbifold version of a conjecture of Maulik and Pandharipande (Topology, 2006) on the Gromov-Witten theory of blowups along complete intersections, a conjecture on the Gromov-Witten theory of root constructions and a conjecture on the Leray-Hirsch result for the orbifold Gromov-Witten theory of Tseng and You (J Pure Appl Algebra, 2016). orbifold Gromov-Witten theory; Leray-Hirsch result; weighted projective bundle; weighted blowup; root stack; blowup along complete intersection Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Topology and geometry of orbifolds Orbifold Gromov-Witten theory of weighted blowups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 determines a large class of examples of a certain invariant Hilbert scheme introduced in [\textit{V.~Alexeev} and \textit{M.~Brion}, J. Algebr. Geom. 14, 83--117 (2005; Zbl 1081.14005)] by relating them to (strict) wonderful varieties, a class of algebraic varieties, equipped with an action of a linear semisimple group and satisfying axioms inspired by the well known compactifications of symmetric spaces of \textit{C. De Concini} and \textit{C. Procesi} [in: Invariant theory, Proc. 1st 1982 Sess. C.I.M.E., Montecatini/Italy, Lect. Notes Math. 996, 1--44 (1983; Zbl 0581.14041)]. Early in the paper, the authors nicely illustrate their main results with a few explicit examples.
We set up some notation. Let \(G\) be a complex connected semisimple algebraic group and let \(\{\lambda_1, \ldots, \lambda_s\}\) be linearly independent dominant weights (after the choice of a Borel subgroup and a maximal torus in it), such that the monoid \(\Gamma\) they generate in the full monoid of dominant weights \(\Lambda^+\) satisfies the following saturation condition due to \textit{D.~Panyushev} [Ann. Inst. Fourier 47, No. 4, 985--1011 (1997; Zbl 0878.14008)]: \(\mathbb{Z}\Gamma \cap \Lambda^+ = \Gamma\). Here \(\mathbb{Z}\Gamma\) stands for the subgroup generated by \(\Gamma\) in the character group of the maximal torus.
The authors show that the invariant Hilbert scheme \(\mathrm{Hilb}^G_{\underline{\lambda}}(V)\) is isomorphic to an affine space (with a specific linear action by the adjoint torus \(T_{\mathrm{ad}}\) of \(G\)). By definition, this Hilbert scheme parametrizes the \textit{nondegenerate} closed \(G\)-subvarieties \(Y\) of \(V:=V(\lambda_1) \oplus \ldots \oplus V(\lambda_s)\) with the property that as \(G\)-modules
\[
\mathbb{C}[Y] \cong \bigoplus_{\lambda \in \Gamma} V(\lambda)^*.
\]
Here \(V(\lambda)^*\) is the dual of the \(G\)-module associated to the dominant weight \(\lambda\) and a subvariety of \(V\) is called nondegenerate if its projection to every isotypical component of \(V\) is nontrivial.
The proof, which uses wonderful varieties, roughly runs as follows. First the authors determine the tangent space \(T_{X_0}\mathrm{Hilb}^G_{\underline{\lambda}}(V)\) to \(\mathrm{Hilb}^G_{\underline{\lambda}}(V)\) at its `most degenerate point' \(X_0\). This \(X_0\) is the affine multicone of the title. To prove that \(\mathrm{Hilb}^G_{\underline{\lambda}}(V)\) is smooth they find another closed point \(X_1\) on \(\mathrm{Hilb}^G_{\underline{\lambda}}(V)\) and, using a result from [\textit{V.~Alexeev} and \textit{M.~Brion}, loc. cit.], they determine the dimension of the (open) \(T_{\mathrm{ad}}\)-orbit on \(\mathrm{Hilb}^G_{\underline{\lambda}}(V)\) through \(X_1\). As this dimension agrees with that of \(T_{X_0}\mathrm{Hilb}^G_{\underline{\lambda}}(V)\) it follows that \(\mathrm{Hilb}^G_{\underline{\lambda}}(V)\) is smooth and, again by [\textit{V.~Alexeev} and \textit{M.~Brion}, loc. cit.], that it is isomorphic to an affine space.
To find the closed point \(X_1\), the authors proceed as follows, generalizing an idea exploited by \textit{S. Jansou} [J. Algebra 306, No. 2, 461--493 (2006; Zbl 1117.14005)]. They first associate a certain combinatorial datum \((S^p(\Gamma),\Sigma(\Gamma))\), introduced by \textit{D. Luna} [Hautes Études Sci. Publ. Math. 91, 161--226 (2001, Zbl 1085.14039)] and called \textit{spherical system}, to \(\Gamma\). (Actually, Luna's spherical systems are more general. Because \(\Gamma\) satisfies Panyushev's saturation condition the spherical systems considered here are \textit{strict}; cf. the appendix of [\textit{D.~Luna}, J. Algebra 313, No. 1, 292--319 (2007; Zbl 1116.22006)]).
Next, they argue that there exists a wonderful variety \(X\) \textit{with} spherical system \((S^p(\Gamma),\Sigma(\Gamma))\), thus confirming a conjecture of Luna [loc.~cit.] for the case of strict spherical systems and strict wonderful varieties. The main part of the proof of the existence of \(X\) is not given in the paper under review; instead the reader is referred to a preprint version [\url{arXiv: math.AG/0603690}]. More recently, the authors expanded this existence proof in [Classification of strict wonderful varieties, \url{arxiv:0806.2263}].
Continuing with the sketch of the proof, the authors also show that \(X\) embeds \(G\)-equivariantly into \(\mathbb{P}(V(\lambda_1)) \times \ldots \times \mathbb{P}(V(\lambda_s))\). The desired subvariety \(X_1\) of \(V\), finally, is the \(G\)-orbit closure of a vector \(v \in V\) which lies in the affine multicone \(\widetilde{X}\) over \(X\) and whose image \([v]\) in \(X\) belongs to the open \(G\)-orbit of \(X\) (wonderful varieties have a unique open orbit). This finishes the overview of the proof.
Furthermore, the authors establish that the universal family over \(\mathrm{Hilb}^G_{\underline{\lambda}}(V)\) is given by the quotient morphism \(\mathrm{Spec}(\mathbb{C}[\pi^{-1}(X)]) \to \pi^{-1}(X)/G\), where \(\pi: \widetilde{X} \dashrightarrow X\) is the natural (rational) map. linear algebraic groups; multicones over flag varieties; spherical varieties; wonderful varieties; equivariant deformations; invariant Hilbert schemes Bravi, P; Cupit-Foutou, S, \textit{equivariant deformations of the affine multicone over a ag variety}, Adv. Math., 217, 2800-2821, (2008) Group actions on varieties or schemes (quotients), Homogeneous spaces and generalizations, Fine and coarse moduli spaces, Representation theory for linear algebraic groups Equivariant deformations of the affine multicone over a flag variety | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Mirror symmetry is a duality between symplectic geometry (A-model) and complex geometry (B-model) for mirror manifolds. In the classical phase (Calabi-Yau phase), the A-model contains information on Gromov-Witten invariants while the B-model contains information on variations of Hodge structures and Bershadsk-Cecotti-Ooguri-Vafa's Kodaira-Spencer theory of gravity. In the Landau-Ginzburg phase, a candidate A-model is FJRW theory constructed by Fan-Jarvis-Ruan based on Witten's idea while the B-model is Saito-Givental's theory of singularities. Ruan proposed a LG/CY correspondence [\textit{Y. Ruan}, ``The Witten equation and the geometry of the Landau-Ginzburg model'', Proc. Symp. Pure Math. 85, 209--240 (2012)] which connects FJRW theory and Gromov-Witten theory via wall-crossing.
Primitive forms (generalizations of differentials of the first kind on elliptic curves) were introduced by \textit{K. Saito} [J. Fac. Sci., Univ. Tokyo, Sect. I A 28, 775--792 (1981; Zbl 0523.32015)]; Publ. Res. Inst. Math. Sci. 19, 1231--1264 (1983; Zbl 0539.58003)] in his study of deformation theory of isolated hypersurface singularities, which are the key ingredients for his theory of singularities. The existence of primitive forms for isolated hypersurface singularities is due to \textit{M. Saito} [Ann. Inst. Fourier 39, No. 1, 27--72 (1989; Zbl 0644.32005)].
The authors considered cusp singularities of the form \(f_{A}(\mathbf{x})=x_{1}^{a_{1}}+x_{2}^{a_{2}}+x_{3}^{a_{3}}+x_{1}x_{2}x_{3}\), (see for instance [\textit{E. J. N. Looijenga}, Isolated singular points on complete intersections. Cambridge etc.: Cambridge University Press (1984; Zbl 0552.14002)]) and determined their primitive forms \(\zeta_{A}\)'s, where \(A=(a_{1},a_{2},a_{3})\) is a triple of positive integers such that \(a_{1}\leq a_{2}\leq a_{3}\) and \(\frac{1}{a_{1}}+\frac{1}{a_{2}}+\frac{1}{a_{3}}<1\). With primitive forms, they described the Frobenius manifold structure on the deformation space of the isolated hypersurface singularity via the standard construction, see [\textit{C. Hertling}, Frobenius manifolds and moduli spaces for singularities. Cambridge: Cambridge University Press (2002; Zbl 1023.14018)]. As a consequence, they obtained a mirror isomorphism of Frobenius manifolds between the one constructed from \((f_{A},\zeta_{A})\) and the one constructed from the genus zero Gromov-Witten theory of the orbifold \(\mathbb{P}^{1}_{A}\). Finally, they also investigated the period mapping of the primitive form. primitive form; Frobenius manifold; cusp singularity; mirror symmetry Y. Shiraishi and A. Takahashi, On the Frobenius manifolds from cusp singularities, arXiv:1308.0105. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Mirror symmetry (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds On the Frobenius manifolds for 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 The aim of this article is to sharpen a theorem by Kempf about deformation of the symmetric product \(C^{(r)}\) of a non-hyperelliptic curve \(C\). The author proves the following:
Theorem: Let \(C\) be a smooth curve of genus \(g\geq 2\), \(r\geq 2\) an integer. Then the natural map \(\text{Def}(C)\to\text{Def}(C^{(r)})\) is an isomorphism if and only if \(g\geq 3\).
To prove this result we have to compare the deformation of the product \(C^r\) with the deformation of the symmetric product \(C^{(r)}\), i.e. the deformation of a smooth variety with the deformation of its quotient by a finite group: the symmetric group \(S_r\).
Lemma: If the genus of \(C\) is at least 2, the natural map \((\text{Def}(C))^r\to\text{Def}(C^r)\) is an isomorphism, it induces an isomorphism \(\text{Def}(C)\to\text{Def}(C^r)^{S_r}\). The functors \(\text{Def}(C^r)\) and \(\text{Def}(C^r)^{S_r}\) are unobstructed. -- By Künneth and by vanishing of \(H^0(C,\theta_C)\), we get \(H^1(C^r,\theta_{C^r})\simeq\bigoplus H^1(C,\theta_C )\). Hence the natural map \((\text{Def}(C))^r\to\text{Def}(C^r)\) is an isomorphism on tangent spaces and, as \((\text{Def}(C))^r\) is unobstructed, it is an isomorphism.
The tangent space of \(\text{Def}(C^r)^{S_r}\) is the \(S_r\)- invariant subspace of \(\bigoplus H^1(C,\theta_C)\), hence the map \(\text{Def}(C)\to\text{Def}(C^r)^{S_r}\) is an isomorphism on tangent spaces and the unobstructness of \(\text{Def}(C)\) implies that it is an isomorphism. Then the theorem is equivalent to: \(\text{Def}(C^r)^{S_r}\to\text{Def}(C^{(r)})\) is an isomorphism. As \(\text{Def}(C^r)^{S_r}\) is unobstructed, it is enough to have an isomorphism on the tangent spaces.
Let \(B\) be the branch locus of the quotient map \(\pi:C^r\to C^{(r)}\), we define the subsheaf \(\theta_{C^{(r)}}(-\log B)\) of derivations of \(\theta_{C^{(r)}}\) respecting the ideal of \(B\) in \(C^{(r)}\) and the equisingular normal sheaf \({\mathcal N}_{B/C^{(r)}}'\) as the quotient \({\mathcal N}_{B / C^{(r)}}' = \theta_{C^{(r)}} / (\theta_{C^{(r)}} (- \log B))\). We have an isomorphism \(\pi_*^{S_r } (\theta_{C^r}) \simeq \theta_{C^{(r)}} (- \log B)\) and the map on tangent spaces of \(\text{Def} (C^r)^{S_r} \to \text{Def} (C^{(r)})\) fits into an exact sequence:
\[
H^0 \bigl( B, {\mathcal N}_{B/C^{(r)} }' \bigr) \to H^1 \bigl( C^r, \theta_{C^r} \bigr)^{S_r} \to H^1 \bigl( C^{(r)}, \theta_{C^{(r)}} \bigr) \to H^1 \bigl( B, {\mathcal N}_{B/C^{(r)}}' \bigr) \to H^2 \bigl( C^r, \theta_{C^r} \bigr)^{S_r}.
\]
For \(i = 0, 1\), \(H^i (B, {\mathcal N}_{B/C^{(r)}}')\) is canonically isomorphic to \(H^i (C,2 \theta_C)\), and by Künneth decomposition we have \(H^2 (C^r, \theta_{C^r}) \simeq \bigoplus_{i \neq j} H^1 (C_i, \theta_{C_i})\otimes H^1(C_j,{\mathcal O}_{C_j})\). Then \(H^1(C^r,\theta_{C^r})^{S_r}\to H^1(C^{(r)},\theta_{C^{(r)}})\) is an isomorphism and we get the theorem. deformation of the symmetric product of a non-hyperelliptic curve; genus B. Fantechi, Déformations of symmetric products of curves.Contemporary Math. 162 (1994), 135--141. Curves in algebraic geometry, Formal methods and deformations in algebraic geometry Deformations of symmetric products of curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper proves some results about three types of DG categories over algebraic stacks \(\mathcal{Y}\): QCoh(\(\mathcal{Y}\)) the DG category of quasi-coherent sheaves on \(\mathcal{Y}\), IndCoh(\(\mathcal{Y}\)) the ind-completion of the full subcategory Coh(\(\mathcal{Y}\)) of QCoh(\(\mathcal{Y}\)) consisting of bounded complexes with coherent cohomology sheaves, and D-mod(\(\mathcal{Y}\)) the DG category of D-modules on \(\mathcal{Y}\). Those DG categories are not independent; results about IndCoh\((\mathcal{Y})\) will follow from those on QCoh\((\mathcal{Y})\), and results about D-mod(\(\mathcal{Y})\) will follow from using some adjoint pair of functors that we have by virtue of the existence of a left adjoint to some conservative forgetful functor D-mod(\(\mathcal{Y}) \rightarrow \text{IndCoh}(\mathcal{Y})\). The DG categories considered in this paper are cocomplete, in which case we can study whether they have compact objects and hence whether they are compactly generated. The algebraic stacks considered are of finite type over a field \(k\) of characteristic zero and are assumed to be such that the automorphism groups of their geometric points are affine. Such stacks are referred to as QCA algebraic stacks. For a prestack \(\mathcal{Y}\) and a morphism \(p_{\mathcal{Y}}: \mathcal{Y} \rightarrow \) pt, \(\Gamma(\mathcal{Y}, -):=(p_{\mathcal{Y}})_*\). By stratifying QCA algebraic stacks \(\mathcal{Y}\) by locally closed substacks, the authors prove that there is an integer \(n\) depending only on \(\mathcal{Y}\) such that \(H^i(\Gamma(\mathcal{Y}, \mathcal{F}))=0\) for all \(i>n\) and all \(\mathcal{F} \in \text{QCoh}(\mathcal{Y})^{\leq 0}\), from which they infer that the functor QCoh(\(\mathcal{Y}) \rightarrow \text{Vect}: \mathcal{F} \mapsto \Gamma(\mathcal{Y}, \mathcal{F})\) is continuous. From that result they further consider those QCA algebraic stacks that in addition have the property of being what they call locally almost of finite type, meaning that they have an atlas with a DG scheme that can be covered by affines Spec(\(A\)) such that \(H^0(A)\) is a finitely generated algebra over \(k\) and each \(H^{-i}(A)\) is a finitely generated \(H^0(A)\)-module. From the above result they prove that for a QCA algebraic stack locally almost of finite type \(\mathcal{Y}\) then IndCoh\((\mathcal{Y})\) and D-mod(\(\mathcal{Y}\)) are compactly generated. There is no such result for QCoh(\(\mathcal{Y}\)) but the authors first prove that the DG category IndCoh\((\mathcal{Y})\) is dualizable, from which they also prove that if in addition the algebraic stack \(\mathcal{Y}\) is eventually coconnective, QCoh(\(\mathcal{Y})\) is dualizable as well. The authors also prove that for \(\mathcal{Y}\) a quasi-compact stack, the functor \(\Gamma_{\text{dR}}(\mathcal{Y}, -)\) is continuous if and only if \(\mathcal{Y}\) is safe, meaning that the neutral connected component of all of its geometric points' automorphism group is unipotent. The authors also fix the problem of having the functor \(\pi_{\text{dR},*}\) of direct image on D-modules not being continuous by defining what they call a renormalized direct image functor \(\pi_{\blacktriangle}\), the dual of \(\pi^!\) where \(\pi: \mathcal{Y}_1 \rightarrow \mathcal{Y}_2\) is a map of QCA algebraic stacks. Such a definition is made possible thanks to a generalization of Verdier duality for D-modules on QCA algebraic stacks locally almost of finite type. Such a renormalized direct image functor is continuous, and they prove that on safe objects we have an isomorphism \(\pi_{\blacktriangle} \rightarrow \pi_{\text{dR},*}\). In addition to these results other peripheral but still very interesting results are presented. For lack of sufficient references in the literature regarding the concepts covered in the paper, the authors liberally provide some highly valuable background material. The exposition does not lack in interesting examples and counter-examples where needed. D-modules; algebraic stacks; DG categories; DG schemes; Verdier duality; quasi-compact stacks; compact generation; de Rham cohomology; ind-completion; quasi-coherent sheaves; symmetric monoidal categories; projection formulas Drinfeld, V.; Gaitsgory, D., \textit{on some finiteness questions for algebraic stacks}, Geom. Funct. Anal., 23, 149-294, (2013) Generalizations (algebraic spaces, stacks), de Rham cohomology and algebraic geometry, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Monoidal categories (= multiplicative categories) [See also 19D23], Derived categories, triangulated categories On some finiteness questions for algebraic 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 The purpose of this paper is to throw a bridge between two seemingly unrelated subjects. One is the Hilbert scheme of points on projective surfaces, which has been intensively studied by various people. The other is the infinite dimensional Heisenberg algebra which is closely related to affine Lie algebras.
We shall construct a representation of the Heisenberg algebra on the homology group of the Hilbert scheme. In other words, the homology group will become a Fock space. The basic idea is to introduce certain ``correspondences'' in the product of the Hilbert scheme. Then they define operators on the homology group by a well-known procedure. They give generators of the Heisenberg algebra, and the only thing we must check is that they satisfy the defining relation. Here we remark that the components of the Hilbert scheme are parameterized by numbers of points and our representation will be constructed on the direct sum of homology groups of all components. Our correspondences live in the product of the different components. Thus it is quite essential to study all components together. -- Our construction has the same spirit as the author's earlier construction [\textit{H. Nakajima}, Duke Math. J. 76, No. 2, 365-416 (1994; Zbl 0826.17026) and Int. Math. Res. Not. 1994, No. 2, 61-74 (1994; Zbl 0832.58007)] of representations of affine Lie algebras on homology groups of moduli spaces of ``instantons'' on ALE spaces which are minimal resolutions of simple singularities. Certain correspondences, called Hecke correspondences, were used to define operators. These twist instantons along curves (irreducible components of the exceptional set), while ours twist ideals around points. In fact, the Hilbert scheme of points can be considered as the moduli space of rank 1 vector bundles, or more precisely torsion free sheaves. Our construction should be considered as a first step to extend the two papers cited above to general 4-manifolds.
Another motivation of our study is the conjecture about the generating function of the Euler number of the moduli spaces of instantons, which was recently proposed by \textit{C. Vafa} and \textit{E. Witten} [Nucl. Phys., B 431, No. 1-2, 3-77 (1994)]. They conjectured that it is a modular form under certain conditions. If \(X^{[n]}\) is the Hilbert scheme parameterizing \(n\)-points in \(X\), then the generating function of the Poincaré polynomials is given by
\[
\sum^\infty_{n= 0} q^nP_t(X^{[n]}) = \prod^\infty_{m=1} {(1+t^{2m-1} q^m)^{b_1 (X)} (1+t^{2m+1} q^m)^{b_3(X)} \over(1-t^{2m-2} q^m)^{b_0(X)} (1-t^{2m}q^m)^{b_2 (X)} (1-t^{2m+2} q^m)^{b_4(X)}},\tag{1}
\]
where \(b_i(X)\) is the Betti number of \(X\).
The paper is organized as follows. In section 2 we give preliminaries. We recall the definition of the convolution product in \S 2(i) with some modifications and describe some properties of the Hilbert scheme \(X^{[n]}\) and the infinite Heisenberg algebra and its representations in \S\S 2(ii), 2(iii). The definition of correspondences and the statement of the main result are given in section 3. The proof will be given in section 4.
While the author was preparing this manuscript, he learned that a similar result was announced by \textit{T. Grojnowski} [Math. Res. Lett. 3, No. 2, 275-291 (1996; Zbl 0879.17011)] who introduced exactly the same correspondence. Heisenberg algebra; Hilbert scheme of points Nakajima, H.: Heisenberg algebra and Hilbert schemes of points on projective surfaces. Ann. Math. (2). \textbf{145}(2), 379-388 (1997). arXiv:alg-geom/9507012. http://dx.doi.org/10.2307/2951818 Parametrization (Chow and Hilbert schemes), Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras Heisenberg algebra and Hilbert schemes of points on projective surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The main goal of this extend paper is to give a geometric explanation of the problem posed by Ginzburg of finding a geometric interpretation of Broer's covariant theorem in the context of geometric Satake. For a simply-connected simple algebraic group \(G\) over \(\mathbb{C}\), the authors show a subvariety of its affine Grassmannian \(\mathbf{Gr}\) that is closely related to the nilpotent cone \(\mathcal{N}\) of \(G\), generalizing a well-known result about \(\mathrm{GL}_n\). By using this subvariety, they construct a sheaf-theoretic functor that leads, when combined with the geometric Satake equivalence and the Springer correspondence, to a geometric explanation for a number of known facts, previously obtained by Broer and Reeder, about small representations of the dual group. As the main result, they prove that there is an action of \(\mathbb{Z} / 2 \mathbb{Z}\) on \(\mathcal{M}\) commuting with the \(G\)-action, and a finite \(G\)-equivariant map \(\mathcal{M} \mapsto \mathcal{N}\) that induces a bijection between \(\mathcal{M} / (\mathbb{Z} / 2 \mathbb{Z})\) and a certain closed subvariety \(\mathcal{N}_{sm}\) of \(\mathcal{N}\), where \(\mathbf{Gr}_{sm}\) is the closed subvariety of \(\mathbf{Gr}\) corresponding to small representation under geometric Satake and \(\mathcal{M} \in \mathbf{Gr}\) is the intersection of \(\mathbf{Gr}_{sm}\) with the opposite Bruhal cell. affine Grassmannian; nilpotent orbits; Springer correspondence Achar, P., Henderson, A.: Geometric Satake, Springer correspondence, and small representations. Preprint. arXiv:1108.4999 (2012) Coadjoint orbits; nilpotent varieties, Group actions on varieties or schemes (quotients), Grassmannians, Schubert varieties, flag manifolds, Representation theory for linear algebraic groups Geometric Satake, Springer correspondence and small 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 setting of this paper is the following: let \(\mathcal X/(0\in\Delta)\) be a one parameter deformation of a complex normal projective surface \(X_0\) with at worst quotient singularities. This deformation is said to be \(\mathbb Q\)-Gorenstein if \(K_\mathcal X\) is \(\mathbb Q\)-Cartier. Locally, a quotient singularity admitting a \(\mathbb Q\)-Gorenstein smoothing is either a rational double point or a cyclic quotient singularity \(\frac{1}{dn^2}(1,dna-1)\) where \(d,n,a>0\) are integers with \(n>a>0\) and \(\gcd(n,a)=1\). The singularity in the latter is called a singularity of class \(T\). In the case \(d=1\), \(\frac{1}{n^2}(1,na-1)\) is called a Wahl singularity.
For a projective normal surface \(X\) with a unique Wahl singularity and satisfying some technical conditions, it is known that a \(\mathbb Q\)-Gorenstein smoothing of \(X\) gives rise to an exceptional vector bundle (a vector bundle \(E\) such that \(\bigoplus_p\mathrm{Ext}^p(E,E)=\mathbb C\)) on the general fiber, see [\textit{P. Hacking}, Duke Math. J. 162, No. 6, 1171--1202 (2013; Zbl 1282.14074)].
The author in this paper studies the natural question that arises considering \(d>1\). The main result shows that the situation generalizes nicely, giving \(d\) exceptional vector bundles which are orthogonal to each other. Let \(X\) be a normal projective surface, \(P\in X\) a singularity of class \(T\) and \(\mathcal X/(0\in\Delta)\) a \(\mathbb Q\)-Gorenstein smoothing of \(X\) (plus some other technical conditions) then, after a finite base change \((0'\in\Delta')\to (0\in\Delta)\), there exist reflexive sheaves \(\mathcal E_1,\dots, \mathcal E_d\) over \(\mathcal X'=\mathcal X\times_\Delta \Delta'\) such that the restrictions to the general fiber \(S\) are exceptional vector bundles \(E_1,\dots,E_d\). Moreover, \(\mathrm{Ext}^p(E_k,E_l)=0\) for each \(p\) and \(k\neq l\).
Key ingredient of the paper is the study of vanishing cycles in \(H_2(S,\mathbb Z)\) combined with a method by Hacking from the paper mentioned above. algebraic surfaces; singularities of class T; exceptional vector bundles; \(\mathbb{Q}\)-Gorenstein smoothing Deformations of singularities, Derived categories of sheaves, dg categories, and related constructions in algebraic geometry Orthogonal exceptional collections from \(\mathbb{Q}\)-Gorenstein degeneration of surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities For a toric Calabi-Yau \(3\)-fold \(X\) the mirror theory lives on a family of curves in \((\mathbb{C}^*)^2\) called the ``mirror curve''. The Eynard-Orantin topological recursion applied to a smooth complex curve produces an infinite tower of free energies \(F_g\) and meromorphic differentials \(W_n^g\). The remodeling conjecture states that the mirror map applied to \(F_g\) and \(W_n^g\) produces the generating functions of the closed and open Gromov-Witten invariants of \(X\) respectively. The open part of the conjecture for \(X=\mathbb{C}^3\) was proved independently by Chen and Zhou. In this note the authors complete the proof for \(\mathbb{C}^3\) by demonstrating that the free energies reproduce the closed invariants given by the celebrated Faber-Pandharipande formula.
The proof relies on methods of Chen and Zhou. The mirror curve to \(\mathbb{C}^3\) is a sphere with \(3\) punctures (pair of pants), and Chen and Zhou expressed its \(W_n^g\) in terms of Hodge integrals. The authors reduce \(F_g\) to Hodge integrals as well, but the computation is more subtle. A subtlety appears also in the relation to matrix models. The Eynard-Orantin recursion applied to the spectral curve of a matrix model exactly reproduces its correlation functions. However, there is a normalization ambiguity in the computation that may lead to discrepancy in the contributions of the constant maps to free energies. The authors show explicitly that this discrepancy is in fact present for the pair of pants, and in contrast to the open part of the conjecture the matrix model machinery can not be used. In conclusion they ask if there is a more direct computation of \(F_g\) that relies on geometry of the pair of pants rather than reduction to Hodge integrals. toric Calabi-Yau; mirror curve; Gromov-Witten invariants; Eynard-Orantin recursion; Hodge integral; spectral curve of a matrix model Bouchard, V; Catuneanu, A; Marchal, O; Sułkowski, P, The remodeling conjecture and the Faber-pandharipande formula, Lett. Math. Phys., 103, 59-77, (2013) 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 The remodeling conjecture and the Faber-Pandharipande 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 From the authors' abstract: ``We start with a discussion on Alexander invariants, and then prove some general results concerning the divisibility of the Alexander polynomials and the supports of the Alexander modules, via Artin's vanishing theorem for perverse sheaves. We conclude with explicit computations of the twisted cohomology following an idea already exploited in the hyperplane arrangement case, which combines the degeneration of the Hodge to the de Rham spectral sequence with the purity of some cohomology groups.''
More precisely the paper is organized as follows. In section 2 the basic definitions are given, introducing the Alexander modules of an affine hypersurface complement. The authors investigate the relations between the first nontrivial Alexander polynomial in one variable and the corresponding Alexander polynomial in several variables. They moreover express the relation between the characteristic varieties defined using the Fitting ideals and the ones defined using the jumping loci of the cohomology with rank one local coefficients. They moreover treat the simplest local situations: the normal crossing case and the case of isolated non-normal crossing singularities.
Section 3 relates the Alexander invariants of an affine hypersurface complement to the ones of its link at infinity, and estimates the support of the Alexander modules of an affine hypersurface complement in terms of local properties of its projective closure.
Section 4 recalls and slightly extends the idea of combining the degeneration of the Hodge to the de Rham spectral sequence with the purity of some cohomology groups (used first by \textit{H. Esnault, V. Schechtman} and \textit{E. Viehweg} [Invent. Math. 109, No. 3, 557--561 (1992; Zbl 0788.32005)] and by \textit{V. Schechtman, H. Terao} and \textit{A. Varchenko} [J. Pure Appl. Algebra 100, No. 1--3, 93--102 (1995; Zbl 0849.32025)]).
In the last section the authors consider the case of the complement of an arbitrary projective hypersurface, treating in particular detail the case when the hypersurface has at most two irreducible components, each of them having only isolated singularities. Alexander invariants; hypersurface complement Dimca A., Maxim L.: Multivariable Alexander invariants of hypersurface complements. Trans. AMS. 359(7), 3505--3528 (2007) math.AT/0506324 Global theory of complex singularities; cohomological properties, Relations with arrangements of hyperplanes, Mixed Hodge theory of singular varieties (complex-analytic aspects), Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects), Hypersurfaces and algebraic geometry, Vanishing theorems in algebraic geometry, Topological properties in algebraic geometry Multivariable Alexander invariants of hypersurface complements | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The quiver Grassmannian is the projective variety of subrepresentations of a finite-dimensional representation of a quiver with a fixed dimension vector.
Quiver Grassmannians occur naturally in different contexts. Fomin and Zelevinsky introduced cluster algebras in 2000. Caldero and Keller used Euler characteristics of quiver Grassmannians for the categorification of acyclic cluster algebras. This was generalized to arbitrary antisymmetric cluster algebras by Derksen, Weyman and Zelevinsky. The quiver Grassmannians play a crucial role in the construction of Ringel-Hall algebras. Moreover, they arise in the study of general representations of quivers by Schofield and in the theory of local models of Shimura varieties. Motivated by this, we study the geometric properties of quiver Grassmannians, their Euler characteristics and Ringel-Hall algebras. This work is divided into three parts.
In the first part of this thesis, we study geometric properties of quiver Grassmannians. In some cases we compute the dimension of this variety, we detect smooth points and we prove semicontinuity of the rank functions and of the dimensions of homomorphism spaces. Moreover, we compare the geometry of quiver Grassmannians with the geometry of the module varieties and we develop tools to decompose the quiver Grassmannian into irreducible components.
In the following we consider some special classes of quiver representations, called string, tree and band modules. There is an important family of finite-dimensional algebras, called string algebras, such that each indecomposable module is either a string or a band module.
In the second part, for the complex field we compute the Euler characteristics of quiver Grassmannians and of quiver flag varieties in the case that the quiver representation is a direct sum of string, tree and band modules. We prove that these Euler characteristics are positive if the corresponding variety is non-empty. This generalizes some results of Cerulli Irelli.
In the third part, we consider the Ringel-Hall algebra of a string algebra. We give a complete combinatorial description of the product of an important subalgebra of the Ringel-Hall algebra.
In covering theory we resemble the results of the last two parts. Euler characteristics; coverings; quiver Grassmannians; flag varieties; representations of quivers; Ringel-Hall algebras; string algebras; band modules; string modules; tree modules Haupt, N.: Euler Characteristic and Geometric Properties of Quiver Grassmannians. Ph.D Thesis, University of Bonn (2011) Representations of quivers and partially ordered sets, Grassmannians, Schubert varieties, flag manifolds, Quantum groups (quantized enveloping algebras) and related deformations, Group actions on affine varieties, Representation theory of lattices Euler characteristics and geometric properties of quiver Grassmannians. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper under review deals with the quantum invariance of genus zero Gromov-Witten invariants, up to analytic continuation under ordinary flops \(f\) from \(X\) to \(X'\) over a non-trivial smooth base. Such flops form the building blocks to connect birational minimal models. This plays an important role in string theory, and also comparing various birational minimal models in higher-dimensional algebraic geometry. The local geometry is encoded in a triple \((S, F, F')\) where \(S\) is a smooth variety and \(F, F'\) are two rank \(r + 1\) vector bundles over \(S\). If \(Z \subset X\) is the \(f\)-exceptional loci, then \(Z\cong \mathbb P(F) \to S\) with fibers spanned by the flopped curves and \(N_{Z/X} = F'\otimes \mathcal O_Z(-1)\), and similar structure holds for the exceptional loci \(Z'\subset X'\). The most studied case of Atiyah flop corresponds to \(S=\) pt and \(r = 1\). An earlier work of the authors of the paper under review constructs a canonical correspondence between the quantum cohomologies of \(X, X'\) by means of the graph closure
\[
\mathcal F = [\overline{\Gamma}_f]_* : QH(X) \to QH(X')
\]
when \(S=\) pt, in which the crucial idea is to interpret \(\mathcal F\)-invariance in terms of analytic continuations in Gromov-Witten theory.
The main result of the paper under reivew is the generalization the quantum invariance above under ordinary flops over a smooth base. The main results of this paper determines the defect of the cup product under the canonical correspondence and show that it is corrected by the small quantum product attached to the extremal ray. Various reductions to the local models are performed in the way by means of degeneration techniques, WDVV equations, topological recursion relations, and divisorial reconstructions. quantum cohomology; ordinary flops; analytic continuations; degeneration formula; reconstructions Lee, Yuan-Pin; Lin, Hui-Wen; Wang, Chin-Lung, Invariance of quantum rings under ordinary flops I: Quantum corrections and reduction to local models, Algebr. Geom., 3, 5, 578-614, (2016) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Minimal model program (Mori theory, extremal rays) Invariance of quantum rings under ordinary flops. I: Quantum corrections and reduction to local models | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Inspired by algebraic geometry over groups, one of the authors of the paper under review, Boris Plotkin, alone and with collaborators, started about 10 years ago to develop algebraic geometry in a significantly more general setting, over algebras of an arbitrary variety \(\Theta\) of universal algebras. This area holds many similarities to classical algebraic geometry but, instead of closely connected with ideal theory of finitely generated polynomial algebras over fields, is closely associated with congruence theories of finitely generated free algebras of those varieties. The group \(\Aut(\Theta^0)\) of automorphisms of the category \(\Theta^0\) of finitely generated free algebras of \(\Theta\) is of great importance.
In the present paper, the authors define semi-inner automorphisms for the categories of free (semi)modules and free Lie modules. Under natural restrictions on the (semi)ring, all automorphisms of \(\Theta^0\) are semi-inner. This holds for the variety \(_R\mathcal M\) of semimodules over an IBN-semiring \(R\). (IBN means invariant basis number, i.e., if the free objects of finite ranks \(m\) and \(n\) are isomorphic, then \(m=n\).) In particular, this is true if the ring \(R\) is Artinian, Noetherian, PI, if \(R\) is a division semiring, etc. The main results are obtained as consequences of a general approach developed in the setting of semiadditive categorical algebra. The paper concludes with an appendix which provides a new easy-to-prove version of a reduction theorem of Mashevitzky, B. Plotkin, and E. Plotkin, used essentially in the present paper. universal algebraic geometry; free modules over Lie algebras; free semimodules over semirings; semi-inner automorphisms; varieties of universal algebras; congruences of finitely generated free algebras; automorphism groups; free Lie modules Katsov, Y.; Lipyanski, R.; Plotkin, B., Automorphisms of categories of free modules, free semimodules, and free Lie modules, Comm. Algebra, 35, 931-952, (2007) Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Rings arising from noncommutative algebraic geometry, Semirings, Module categories in associative algebras, Identities, free Lie (super)algebras, Automorphisms and endomorphisms of algebraic structures, Categories of algebras, Noncommutative algebraic geometry Automorphisms of categories of free modules, free semimodules, and free Lie modules. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(k\) be an arbitrary algebraically closed field. For any finite subgroup \(G\) of \(\text{SL}(3,k)\) of order prime to the characteristic of the ground field \(k\) the quotient space \(\mathbb{A}^3_k/G\) is then a normal Gorenstein variety with trivial canonical sheaf. In birational geometry, a natural choice of a crepant smooth resolution of the singular variety \(\mathbb{A}^3_k/G\) would be very convenient, but it seems that the general theories in this framework, such as the theory of minimal models and the theory of torus embeddings, do not canonically provide a resolution of this type.
In the paper under review, the author approaches this problem by studying a particular Hilbert scheme \(\text{Hilb}^G(\mathbb{A}^3_k)\), which finally turns out to be a canonical crepant resolution of the quotient variety \(\mathbb{A}^3_k/G\). The so-called \(G\)-orbit Hilbert scheme \(\text{Hilb}^G:= \text{Hilb}^G(\mathbb{A}^3_k)\) is, by definition, the scheme parametrizing all \(G\)-invariant smoothable \(0\)-dimensional subschemes of \(\mathbb{A}^3_k\) of length \(n:=|G|\). This object, which may be regarded as a certain substitute for the quotient \(\mathbb{A}^3_k/G\) was introduced by the author and \textit{Y. Itô} in [Proc. Japan Acad., Ser. A 72, No. 7, 135--138 (1996; Zbl 0881.14002)] as a suitable tool in the study of resolutions of quotient singularities via the famous McKay correspondence.
In the present paper, the \(G\)-orbit Hilbert scheme \(\text{Hilb}^G\) is described for a finite abelian subgroup \(G\) of \(\text{SL}(3,k)\) resulting in the fact that \(\text{Hilb}^G\) appears then as a smooth torus embedding associated to a crepant fan in \(\mathbb{R}^3\) with apices constructed from the group \(G\). Furthermore, it is shown that the commutativity of \(G\) implies the nonsingularity of that associated fan.
This finally establishes the author's main theorem (Theorem 0.1.) stating the following: For any abelian subgroup \(G\) of \(\text{SL}(3,k)\) of order prime to the characteristic of the ground field \(k\) the \(G\)-orbit Hilbert scheme \(\text{Hilb}^G(\mathbb{A}^3_k)\) is a crepant resolution of the quotient space \(\mathbb{A}^3_k/G\).
The first half of the present article is devoted to describing a \(G\)-orbit Hilbert scheme as a toric variety in arbitrary dimension. This part is based on a fine analysis of the corresponding lattices and \(G\)-graphes, which is highly interesting and important for its own sake. In the second half of the paper, the particular case of dimension three and an abelian subgroup \(G\) of \(\text{SL}(3,k)\) is inspected more closely by means of the special appearing \(G\)-graphs, culminating in the author's main theorem mentioned above. At the end of the article, the author discusses some interesting examples in dimension three and four, thereby illustrating the variety of possibilities that can occur already in those low-dimensional cases.
In a sense, the present work may be regarded as a complement to the related earlier results by \textit{Y. Itô} and \textit{M. Reid} [in: Higher-dimensional complex varieties. Proc. Int. Conf. Trento, Italy, June 15--24, 1994. 221--240 (1996; Zbl 0894.14024)] and by \textit{Y. Itô} and \textit{H. Nakajima} [Topology 39, 1155--1191 (2000; Zbl 0995.14001)]. quotient varieties; quotient singularities; resolution of singularities; toric varieties I. Nakamura, \textit{Hilbert schemes of abelian group orbits}, J. Algebraic Geom. \textbf{10} (2001), no. 4, 757-779. Parametrization (Chow and Hilbert schemes), Homogeneous spaces and generalizations, Group actions on varieties or schemes (quotients), Global theory and resolution of singularities (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies Hilbert schemes of abelian group orbits | 1 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper provides a generalization of the result of \textit{A. Bayer} and \textit{E. Macrì} [J. Am. Math. Soc. 27, No. 3, 707--752 (2014; Zbl 1314.14020)] relating the Bridgeland stability manifold and the movable cone of the moduli space of stable objects on smooth projective varieties to the setting of arbitrary separated schemes of finite type.
One of the key ideas is, as stated in the section 1.2, that one should introduce a proper definition of the stability function. For a separated scheme \(Y\) of finite type, the authors define a variant \(K^{\text{num}}_c(Y)\) of the numerical Grothendieck group as the quotient of \(K(D_c(Y))\) by the radical of the Euler pairing with perfect complexes on \(Y\), where \(D_c(Y)\) is the full subcategory of objects with proper support in the bounded derived category. Then they introduce the notion of numerical Bridgeland stability condition for compact support on \(Y\) as a pair \(\sigma=(Z_{\sigma}, \mathcal{P}_{\sigma})\) of a group homomorphism \(Z_{\sigma}:K^{\text{num}}_c(Y) \to \mathbb{C}\) and a slicing \(\mathcal{P}_{\sigma}\) of \(D_c(Y)\).
The main statement is Theorem 1.2.1, where a family of nef divisors on moduli spaces of stable objects in the above sense is constructed. In the proof, the authors construct a family of t-structures with left-compact support, which is introduced in Definition 2.1.3 as a replacement of the ordinary compact support property so that it behaves well under the derived restriction.
This paper also constructs stability conditions in the case when \(Y\) is a smooth scheme, projective over an affine scheme, equipped with a tilting bundle \(E\). In this setting, denoting by \(A\) the endomorphism algebra of \(E^{\vee}\), we have stability conditions \(\sigma_\theta\) on the derived category of finite \(A\)-modules parametrized by stability parameter \(\theta\) for \(A\)-modules in the sense of King. By the tilting equivalence one can construct stability conditions on \(D_c(Y)\) from \(\sigma_\theta\). Then Theorem 1.4.1 says that the numerical divisor class constructed by Theorem 1.2.1 is equivalent to the polarizing ample line bundle on the moduli space of stable \(A\)-modules.
The main result of this paper seems to have a wide range of applications as indicated in the section 1.5. The reviewer recommends this paper especially for those interested in some application of Bridgeland stability conditions to birational geometry of moduli spaces and geometric representation theory. Bridgeland stability conditions; derived categories; t-structures; moduli spaces of sheaves and complexes; nef divisors Bayer, A., Craw, A., Zhang, Z.: Nef divisors for moduli spaces of complexes with compact support. Sel. Math. (N.S.) \textbf{23}(2), 1507-1561 (2017) Algebraic moduli problems, moduli of vector bundles, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], \(K3\) surfaces and Enriques surfaces, Derived categories, triangulated categories Nef divisors for moduli spaces of complexes with compact support | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be a complex reductive group and \(\Gamma\) a finitely generated group. In this paper, the authors introduce techniques for calculating the E-polynomials of the character variety \(\text{Rep}(\Gamma,G)\), which is defined as the GIT quotient of the affine variety \(\text{Hom}(\Gamma,G(\mathbb{C}))\) by the conjugation action of \(G(\mathbb{C})\). The approach is an extension to the singular (untwisted) case of the techniques applied by Hausel and Rodriguez-Villegas to the non-singular (twisted) case.
The central theorem (Theorem 1.1/3.6) is as follows. Suppose that there is a polynomial \(A(t)\in\mathbb{C}(t)\) and a positive integer \(N\) such that, for every finite field \(\mathbb{F}_q\) of order \(q\) with \(q\equiv 1\bmod N\), the number of isomorphism classes of \(n\)-dimensional reductive representations over \(\mathbb{F}_q\) (respectively, the number of isomorphism classes of \(n\)-dimensional reductive representations over \(\mathbb{F}_q\) with trivial determinant) is \(A(q)\). Then \(A(q)\) is the E-polynomial of \(\text{Rep}(\Gamma,\text{GL}_n(\mathbb{C}))\) (respectively \(\text{Rep}(\Gamma,\text{SL}_n(\mathbb{C}))\)). This theorem reduces the computation of the E-polynomial to counting representations. The authors apply the theorem to the cases where \(\Gamma\) is a free group (and \(n=2\)), the fundamental group of a compact oriented surface (\(n=2,3\)) and the fundamental group of a compact non-orientable surface (\(n=2\)). For \(n=2\), these results are already known for \(\Gamma\) a free group and in the case of a compact oriented surface, but the computations here are simpler. character variety; reductive group; representation; E-polynomial Baraglia, D.; Hekmati, P., Arithmetic of singular character varieties and their \textit{E}-polynomials, Proc. Lond. Math. Soc., 20, 3, 293-332, (2017) Algebraic moduli problems, moduli of vector bundles, Mixed Hodge theory of singular varieties (complex-analytic aspects), Ordinary representations and characters, Geometric invariant theory Arithmetic of singular character varieties and their E-polynomials | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(k\) be a field of characteristic zero and let \(X := \mathrm{Rep}_\alpha(Q)\) be the variety of \(k\)-representations with dimension vector \(\alpha\) of a quiver \(Q\). For a further dimension vector \(\gamma\) let \(Y := \mathrm{Rep}_{\gamma \hookrightarrow \alpha}(Q)\) be the subvariety of \(X\) consisting of all \(M \in X\) which have a \(\gamma\)-dimensional subrepresentation. If \(Q\) is the Dynkin \(A_2\)-quiver, then \(Y\) is a usual determinantal variety, whence the name ``quiver determinantal variety'' for \(Y\).
Let \(V := \mathrm{Gr}{\alpha \choose \gamma}\) and let \(Z\) be the (smooth) subvariety of \(V \times X\) consisting of all pairs \((L,M)\) where \(L\) is a subrepresentation of \(M\). The variety \(Z\) is in fact the total space of a subbundle of the (trivial) vector bundle \(V \times X\) over \(V\). Let \(q:V \times X \rightarrow X\) be the projection. Then \(Y = q(Z)\) and we are thus in the setting of Weyman's abstraction \textit{J. M. Weyman} [Cohomology of vector bundles and syzygies. Cambridge Tracts in Mathematics 149. Cambridge: Cambridge University Press (2003; Zbl 1075.13007)] of Lascoux's construction [\textit{A. Lascoux}, Adv. Math. 30, 202--237 (1978; Zbl 0394.14022)] of a minimal free resolution of determinantal varieties. In this theory the pushforward along \(q\) of a Koszul complex on \(V \times X\) twisted by a chosen vector bundle \(\mathcal{V}\) on \(V\) is considered. This is the so-called Kempf--Lascoux--Weyman complex \(F_\bullet(\mathcal{V})\). If the restriction \(q':Z \rightarrow Y\) of \(q\) is birational (a criterion is given in the paper under review) and \(Y\) has rational singularities, then \(F_\bullet(k)\) for the trivial line bundle is a minimal free resolution of \(k [ Y ]\) as a \(k[ X ]\)-module. This nice situation rarely occurs, however (some counter-examples are given in the paper). In the paper the complex \(F_\bullet(\mathcal{L})\) is studied for line bundles \(\mathcal{L}\) on \(V\) in the case of \(Q\) being a Kronecker quiver. The terms of \(F_\bullet(\mathcal{L})\) are determined and a condition is provided when it resolves a maximal Cohen--Macaulay module supported on \(Y\). Furthermore, a vanishing condition for Kronecker coefficients is deduced. In case \(Y\) is of codimension one in \(X\) and \(q\) is birational, the determinant of \(F_\bullet(\mathcal{L})\) is determined. determinantal variety; free resolution; quiver representation; Cohen-Macaulay module; Kronecker coefficient Syzygies, resolutions, complexes and commutative rings, Determinantal varieties, Representations of quivers and partially ordered sets On some quiver determinantal varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The aim of the first part of the article is to construct an analog for triple coverings of Horikawa's resolution for double coverings.
Let \(W\) be a nonsingular complex analytic surface and \(\pi : \overline L \to W\) the \(\mathbb{P}^ 1\)-bundle associated to a line bundle \(L\) on \(W\). Let \(S\) be an irreducible reduced divisor on \(\overline L\) linearly equivalent to \(3T\), where \(T = {\mathcal O}_{\overline L} (1)\). Then we get a triple covering \(\varphi : S \to W\), we call \((S,W,L)\) a ``triple section surface'', and we want to construct a reduction process of singularities of the surface \(S\). There are two classes of singular points \(Q\) on \(S\): the ``inner double points'', for which \(\# \varphi^{ - 1} (\varphi (Q)) = 2\), and the ``target singularities'', for which \(\# \varphi^{ - 1} (\varphi (Q)) = 3\), and in this case we call \(P = \varphi (Q)\) a ``target point'' of \(W\). If \(\tau_ 1 : W_ 1 \to W\) is the blow-up of a target point \(P\) of \(W\), we can construct a new triple section surface \((S_ 1, W_ 1, L_ 1)\), where the line bundle \(L_ 1\) depends on \(L\) and on \(P\), such that we have a birational morphism \(\overline \tau_ 1 : \overline S_ 1 \to S\). We call this a ``triplet blow-up'' at \(P\), and we get the following result:
Theorem: Let \((S,W,L)\) be a triple section surface with \(S\) normal and
\[
(S,W,L) \gets (S_ 1, W_ 1, L_ 1) \gets \cdots \gets (S_ r, W_ r, L_ r) \gets \cdots
\]
be the reduction process by successive triplet blow-ups. Then this process terminates in finite steps, namely there exists \(r\) such that \((S_ r, W_ r, L_ r)\) has no target point.
\(\tau : (S_ r, W_ r, L_ r) \to (S,W,L)\) is called the ``canonical reduction'' and then the author studies the singularities of the (not necessarily normal) surface \(S_ r\) and the exceptional divisor of the resolution of \(S_ r\). He shows that the singularities are only relative cusps, relative nodes and isolated inner double points, all of multiplicity at most 2.
In the second part the author solves Durfee's conjecture for two- dimensional hypersurface singularities of multiplicity 3. If \((V,p)\) is a such singularity, there exists a triple section surface \((S,W,L)\) and a target singularity \(P\) on \(S\) such that \((V,p)\) and \((S,P)\) are analytically equivalent. We can use the results of the first part of the article and we get:
Theorem: Let \((V,p)\) be a normal two-dimensional hypersurface singularity of multiplicity 3. Then we have: \(\mu (V,p) \geq 6p_ g (V,p) + 2\), where \(\mu (V,p)\) is the Milnor number and \(p_ g (V,P)\) the geometric genus of the singularity. -- Especially the signature of the Milnor fiber of \((V,p)\) is negative.
Moreover, the equality holds if and only if \((V,p)\) is a simple elliptic singularity of type \(\widetilde E_ 6\) in the sense of Saito. resolution of surface singularities; triple section surface; target point; triple covering; inner double points; Durfee's conjecture; two- dimensional hypersurface singularities of multiplicity 3; Milnor number; simple elliptic singularity Némethi, A.: Dedekind sums and the signature of \(f(x,y)+z^N\). Selecta. Math. (N.S.) \textbf{4}(2), 361-376 (1998) Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry, Coverings in algebraic geometry, Complex surface and hypersurface singularities, Modifications; resolution of singularities (complex-analytic aspects) Normal two-dimensional hypersurface triple points and the Horikawa type 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 In this paper, the authors study symplectic blowing down.
This is the opposite process of symplectic blowing up which in dimension six results in either a symplectic \(\mathbb{P}^2\) with normal degree \(-1\) or a symplectic \(\mathbb{P}^1\)-bundle over a surface \(\Sigma\) with normal degree \(-1\) along the \(\mathbb{P}^1\)-fibers. In general, the blowing up construction gives rise to a symplectic exceptional divisor which is known to be a topological exceptional divisor.
A topological exceptional divisor is a codimension-\(2\) submanifold \(D^{2n} \subset (M^{2n+2}, \omega)\) which admits a linear \(\mathbb{P}^k\)-bundle structure \(\pi: D^{2n} \to Y^{2n-2k}\) over an oriented \((2n - 2k)\)-manifold \(Y\) such that (i) the normal line bundle \(N_D\) is the tautological line bundle when restricted to the projective space fibers of \(\pi\), and (ii) \(\omega|_D\) is almost standard.
A matching triple \((X,D', S; \Omega)\) for a topological exceptional divisor \(D\) is a linear \((\mathbb{P}^{k+1}, \mathbb{P}^k, \mathbb{P}^0)\) bundle triple \((X,D', S)\) over \(Y\) with \(\Omega\) being a symplectic form on \(X\), satisfying certain compatibility conditions. Section~2 proves that if a topological exceptional divisor \(D \subset (M, \omega)\) has a matching triple, then up to integral deformation, \((M, \omega)\) can be symplectically blown down along \(D\).
In Section~3, motivated by this observation, the authors investigate matching triples for a topological exceptional divisor \(D \subset (M, \omega)\) with \(\pi: D \to Y = \Sigma\) and \(\Sigma\) being a surface, and define that a topological exceptional divisor \(\pi: D \to \Sigma\) is admissible if the symplectic form \(\omega|_D\) on \(D\) satisfies certain ratio bound depending on the genus of \(\Sigma\). Moreover, the symplectic cone of a linear \((\mathbb{P}^{k+1}, \mathbb{P}^k, \mathbb{P}^0)\) bundle triple over \(\Sigma\) is analyzed. Section~4 verifies that if a symplectic divisor \(\pi: D \to \Sigma\) of \((M, \omega)\) arises from symplectically blowing up a symplectic surface \(\Sigma\) in a symplectic manifold, then it is an admissible topological exceptional divisor. Conversely, if \(\pi: D \to \Sigma\) is an admissible topological exceptional divisor, then there exists a weak matching triple. Applying these results to dimension six, the authors confirm that for a topological exceptional divisor \(\pi: D \to Y\) in a \(6\)-dimensional symplectic manifold \((M, \omega)\), if \(Y\) is a point, then \((M, \omega)\) can be blown down along \(D\).
Furthermore, if \(Y = \Sigma\) is a surface and \(\pi: D \to \Sigma\) is admissible, then a weak matching triple is actually a matching triple and hence, up to integral deformation, \((M, \omega)\) can be blown down along \(D\). For this last result, the paper ends with an explicit and stronger formulation. symplectic birational geometry; symplectic blowing down; symplectic blowing up; symplectic exceptional divisor; topological exceptional divisor Rational and birational maps, Global theory of symplectic and contact manifolds, Holomorphic symplectic varieties, hyper-Kähler varieties, Symplectic and contact topology in high or arbitrary dimension Symplectic blowing down in dimension six | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper is a survey of results based on the observation that there is a bijection between the set of smooth hyperelliptic curves of genus \(g\) and the set of nondegenerate pencils of quadrics in the affine space of dimension \(2g+2\). Beautiful theorems of , e.g., \textit{P. E. Newstead} [Topology 6, 241--262 (1967; Zbl 0201.23401); ibid. 7, 205--215 (1968; Zbl 0174.52901)], \textit{M. S. Narasimhan} and \textit{S. Ramanan} [Ann. Math. (2) 89, 14--51 (1969; Zbl 0186.54902)], \textit{S. Ramanan} [in: Proc. int. Congr. Math., Helsinki 1978, Vol. 2, 543--547 (1980; Zbl 0438.14014)], \textit{U. V. Desale} and \textit{S. Ramanan} [Invent. Math. 38, 161--185 (1976; Zbl 0323.14012)], \textit{U. N. Bhosle} [Compos. Math. 51, 15--40 (1984; Zbl 0539.14011); J. Reine Angew. Math. 407, 75--98 (1990; Zbl 0693.14015)] relate the geometry of hyperelliptic curves (moduli of vector bundles over them) and the geometry of pencils of quadrics (certain spaces constructed starting from them). The survey continues with the case when the curves are allowed to have nodal singularities and the pencils of quadrics are degenerate (no bijection in this case!). Theorems from several papers of the author are stated and some proofs are sketched. pencils of quadrics; (nodal) hyperelliptic curve; vector bundle; Jacobian; torsionfree sheaves Vector bundles on curves and their moduli, Algebraic moduli problems, moduli of vector bundles, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Pencils of quadrics and vector bundles on hyperelliptic curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities If \(G\) is a polyhedral group, then Nakamura's \(G\)-Hilbert scheme \(Y=G\)-Hilb\((\mathbb{C}^3)\) gives a natural Calabi-Yau resolution of the quotient singularity \(\mathbb{C}^3/G\). The authors describe the quantum geometry of \(Y\) in terms of \(R\), an ADE root system associated to \(G\). They give an explicit formula for the Gromov-Witten partition function of \(Y\) and a prediction for the orbifold Gromov-Witten invariants of the quotient singularity \(\mathbb{C}^3/G\), via the Crepant Resolution Conjecture. polyhedral group; \(G\)-Hilbert scheme; Calabi-Yau resolution; Gromov-Witten invariants Jim Bryan and Amin Gholampour, The quantum McKay correspondence for polyhedral singularities, Invent. Math. 178 (2009), no. 3, 655-681. Global theory and resolution of singularities (algebro-geometric aspects), Calabi-Yau manifolds (algebro-geometric aspects), Modifications; resolution of singularities (complex-analytic aspects) The quantum McKay correspondence for polyhedral 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 regularized volume for hyper-Kähler quotients is introduced, and regularized volumes of the Hitchin space, the moduli space of instantons and quiver varieties are computed. It is shown in the case of Hitchin spaces that the evaluation of the volume reduces to a summation over solutions of Bethe Ansatz equations for the nonlinear Schrödinger system.
To define the regularized volume, first Kähler and hyper-Kähler quotients are explained in Section 2. Let \(X\) be a symplectic manifold with the symplectic form \(\omega\), \(K\) a compact group acting on \(X\) and preserving \(\omega\), \(\mu:X\to \underline{\mathfrak k}^*\) be an equivariant momentum map: \(d\langle \mu,\xi \rangle= -\iota_{V_\xi} \omega\). The symplectic quotient is \(X//K= \mu^{-1}(0)/K\). If \(X\) is a Kähler manifold, then \(X//K\) is a Kähler manifold. If \(X\) is a hyper-Kähler manifold and \(K\) preserves the hyper-Kähler structure, then the momentum map is extended to the hyper-Kähler moment map \(\vec\mu: X\to \underline{\mathfrak k}^*\otimes \mathbb{R}^3\), \(d\langle\vec \mu,\xi\rangle =-\iota_{V_\xi} \vec \omega\), where \(\vec\omega\) is the triplet of symplectic forms. The hyper-Kähler quotient is \(X////K= \vec\mu^{-1} (0)/K\). It is a hyper-Kähler manifold [\textit{N. Hitchin}, \textit{A. Karlhede}, \textit{U. Lindström} and \textit{M. Roček}, Commun. Math. Phys. 108, 535-589 (1987; Zbl 0612.53043)]. By using these notations, the following definitions are given:
(a) The Hamiltonian regularized volume: \(\text{Vol}_\varepsilon ({\mathcal M})= \int_{\mathcal M}e^{ \overline \omega-H_\varepsilon}\). Here, \({\mathcal M}\) is acted on by a group \(H\) in a Hamiltonian way such that for some \(\varepsilon\in \underline h\) the Hamiltonian \(H_\varepsilon= \langle\mu_h, \varepsilon \rangle\) is sufficiently positive at infinity.
(b) The hyper-Kähler regularized volume: \(\text{Vol}_{ \vec \varepsilon} ({\mathcal M}(\vec\zeta))=\int_{{\mathcal M}(\vec\zeta)} \text{vol}_g e^{\langle \vec\varepsilon, \vec\mu_t \rangle}\).
The authors remark that this definition is the direct analogue of the ``equivariant volumes'' of \textit{A. B. Givental} [Equivariant Gromov-Witten invariants, Int. Math. Res. Not. 1996, 613-663 (1996; Zbl 0881.55006)]. But it requires a triholomorphic action on \({\mathcal M}\) of the torus of dimension \(\dim{\mathcal M}/4\). Let \({\mathfrak T}\) be such a torus and \(\vec\varepsilon \in{\mathfrak t}\otimes \mathbb{R}^3\). If \(\vec \varepsilon \in{\mathfrak t}\otimes \mathbb{R}\), then this regularization reduces to the Hamiltonian regularization (Section 3).
To compute the regularized volume, localization is useful. It is explained in Section 2. In Section 4, the regularized volume of ALE gravitational instanton is computed in terms of the fundamental weights in the \(A_n\) case. The regularized volume of the Hitchin space is computed in Section 5. It is expressed as the double summand with respect to the decomposition of the bundle as the sum of line bundles and the decomposition of the Chern numbers. So there are two ways to get the answer and this leads to the relation of the Bethe Ansatz equation for the nonlinear system and the regularized volume. Volumes of the moduli spaces of instantons and quiver varieties are computed in Sections 6 and 7. regularized volume; hyper-Kähler quotients; symplectic manifold; equivariant momentum map; symplectic quotient; Hamiltonian regularization D. Xie and S.-T. Yau, \textit{New}\( \mathcal{N} \) = 2 \textit{dualities}, arXiv:1602.03529 [INSPIRE]. Momentum maps; symplectic reduction, Supersymmetric field theories in quantum mechanics, Applications of global differential geometry to the sciences, String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Hyper-Kähler and quaternionic Kähler geometry, ``special'' geometry, Group actions and symmetry properties, Algebraic moduli problems, moduli of vector bundles Integrating over Higgs branches | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper studies the topology of highly nonisolated singularities which arise as the hypersurface of exceptional orbits for a representation of a complex linear algebraic group with open orbit. In particular, the determinantal hypersurface in the space of \(m\times m\) (symmetric, skew symmetric) matrices is studied, and the case of equidimensional representations \(\rho: G \to\mathrm{GL}(V)\), meaning that \(\dim G=\dim V\). This includes certain quiver representations. The exceptional orbit variety is a linear free (or sometimes slightly weaker a linear free\(^*\)) divisor. The defining equation \(f\) is homogeneous, so as Milnor fiber one can take the global affine hypersurface \(f^{-1}(1)\). Starting from the topology of the complement, the cohomology of the Milnor fiber and the link are determined. For the cases of general \(m\times m\) matrices the Milnor fiber is \(\mathrm{SL}_m(\mathbb C)\), for symmetric matrices \(\mathrm{SL}_m(\mathbb C)/\mathrm{SO}_m(\mathbb C)\). From this follows that the cohomology of the Milnor fibers and complements are isomorphic as algebras to exterior algebras or for one family, modules over exterior algebras.
In the final section, Thom-Sebastiani sums of these types of hypersurfaces are studied, and also sums with weighted homogeneous nonisolated singularities with Milnor fiber a bouquet of spheres. The resulting Milnor fibers are bouquets of spaces, each of which are suspensions of joins of compact manifolds. nonisolated singularities; prehomogeneous vector spaces; determinantal varieties; Milnor fiber Complex surface and hypersurface singularities, Discriminantal varieties and configuration spaces in algebraic topology, Prehomogeneous vector spaces, Homology and cohomology of homogeneous spaces of Lie groups, Determinantal varieties Topology of exceptional orbit hypersurfaces of prehomogeneous 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 Let \(SU_ X (r)\) be the moduli space of semistable rank-\(r\) vector bundles over a compact Riemann surface \(X\) of genus \(g\), whose determinant bundle is trivial. There is a natural polarizing line bundle \(L\) over \(SU_ X (r)\), the so-called generalized theta bundle. The \(\mathbb{C}\)-vector space of sections of \(L^{\otimes k}\), \(k \in \mathbb{N}\), is called the space of generalized theta functions of order \(k\) on \(SU_ X (r)\), and its dimension may be computed by the famous Verlinde formula.
The Verlinde formula was first discovered by physicists, in the context of conformal quantum field theory [cf. \textit{E. Verlinde}, ``Fusion rules and modular transformations in \(2d\) conformal field theory'', Nucl. Phys. B 300, No. 3, 360-376 (1988)], served then as a conjectural problem (even for more general semistable vector bundles on curves) in algebraic geometry for some years, and was recently affirmatively established, partly in special cases, by several authors [A. Bertram -- A. Szenes (1991), M. Thaddeus (1992), G. Faltings (1993)].
The aim of the present paper is to give a proof of the Verlinde formula, in the case mentioned above, by explicitly relating the constructions and arguments of the physicists to computing \(H^ 0 (SU_ X(r), L^{\otimes k})\). This is done by establishing a canonical isomorphism between this space and the so-called space of conformal blocks of level \(k\), denoted by \(B_ k (r)\), which naturally arises in the representation theory of Kac-Moody algebras and their applications in conformal quantum field theory. Having constructed this canonical isomorphism in a mathematically rigorous way, the authors derive the Verlinde formula from the dimension formula for \(B_ k (r)\), which has been computed, in a purely combinatorial manner, by \textit{A. Tsuchiya}, \textit{K. Ueno} and \textit{Y. Yamada} [in Integrable systems in quantum field theory and statistical mechanics, Proc. Sympos. 1988, Adv. Stud. Pure Math. 19, 459-566 (1989; Zbl 0696.17010)], and also by \textit{D. Gepner} [Commun. Math. Phys. 141, 381-411 (1991; Zbl 0752.17033)].
At the end of the paper, the results are generalized to the case of semistable vector bundles of arbitrary degree and determinant \({\mathcal O}_ X (dp)\), \(d\) being a fixed integer and \(p \in X\). moduli space; compact Riemann surface; generalized theta bundle; Verlinde formula; conformal quantum field; representation theory of Kac-Moody algebras A. Beauville and Y. Laszlo, ''Conformal blocks and generalized theta functions,'' Comm. Math. Phys., vol. 164, iss. 2, pp. 385-419, 1994. Theta functions and curves; Schottky problem, Vector bundles on curves and their moduli, Quantum field theory on curved space or space-time backgrounds, Theta functions and abelian varieties, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Riemann surfaces; Weierstrass points; gap sequences Conformal blocks and generalized theta 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 This is the first of a series of two papers laying the foundations for the study of algebraic families of Harish-Chandra pairs and the attached modules. See [\textit{J. Bernstein} et al., Int. Math. Res. Not. 2020, No. 11, 3494--3520 (2020; Zbl 1484.22010)] for the second paper in the series.
The authors introduce a general algebraic framework, using sheaf theory, to study some phenomena that occur in relation with a \textit{variation of parameters} in either (a) the structure of a reductive Lie group, or (b) the structure of a representation of a given reductive group, or (c) both at the same time.
This is inspired by the notion of `contraction' from mathematical physics; see [\textit{E. Inönü} and \textit{E. P. Wigner}, Proc. Natl. Acad. Sci. USA 39, 510--524 (1953; Zbl 0050.02601)] and [\textit{I. E. Segal}, Duke Math. J. 18, 221--265 (1951; Zbl 0045.38601)]. In particular, the notion allows for variations of reductive group structures where non-reductive groups appear at singular parameter values.
In the second paper of the series [loc. cit.], the authors prove that their purely algebraic framework naturally produces families of unitary representations, thereby connecting several classical families from mathematical physics which may appear unrelated at first sight.
Section 2 introduces the general framework. Let \(X\) be a complex algebraic variety. Then the authors introduce notions of \textit{algebraic family over} \(X\) for several familiar Lie-theoretic and representation-theoretic objects. They assume \(X\) is an irreducible, nonsingular, quasi-projective complex algebraic variety, and define algebraic families over \(X\) of Lie algebras, algebraic groups, Harish-Chandra pairs \((\mathfrak{g},K)\), of Harish-Chandra modules... The language is that of algebraic geometry: all objects introduced by the authors the authors are sheaves on \(X\), and the variation of objects over \(X\) is encoded by algebraic notions.
The two main examples, given in Sections 2.1.2 and 2.1.3, are over \(X=\mathbb{C}\). Both are based upon the Inonu-Wigner contraction of Lie algebras, and to the deformation to the normal cone construction in algebraic geometry.\\
The other two sections focus on examples. A short Section 3 outlines a construction of algebraic families of real algebraic groups, and specializes to the classical groups. The more detailed Section 4 focuses on a particular family of Harish-Chandra pairs, related to the contraction which connects \(\mathrm{SL}(2,\mathbb{R})\), \(\mathrm{SU}(2)\) and \(\mathrm{SO}(2) \ltimes \mathbb{R}^2\). The main theme is representation theory: the authors outline a classification for the corresponding ``generically irreducible'' families of Harish-Chandra modules. Along the way, they introduce natural invariants for such families of Harish-Chandra modules, which are (new, and nontrivial) family versions of some of the usual representation-theoretic invariants: infinitesimal characters, Casimir eigenvalues, \(K\)-types. contractions of Lie groups; Harish-Chandra modules; algebraic families Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.), Linear algebraic groups over the reals, the complexes, the quaternions, Group schemes Algebraic families of Harish-Chandra pairs | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Fix a non-singular curve \(C\), and an effective divisor \(E\) on \(C\) such that all points on \(E\) are of multiplicity \(1\). Define \(\text{{Sh}}(C,E,Z)\) to be the set of all isomorphism classes of projective varieties \(\pi: \mathcal Z\to C\) with a fiber of ``type'' \(Z\) such that singular fibers occur only over \(E\). The general conjecture of Shafarevich asks:
For which ``type'' of varieties \(Z\) and data \(C,E)\), is \(\text{{Sh}}(C,E,Z)\) finite?
This paper studies an analogue of the Shafarevich conjecture, i.e., the finiteness of the families of polarized Calabi-Yau manifolds over a fixed Riemann surface with prescribed points of degenerate fibers, up to isomorphism. The finiteness is reduced to prove that the moduli space of maps of a Riemann surface to the moduli space of polarized Calabi-Yau manifolds with some additional properties is a finite set.
The rigidity property plays an important role in their proof of the finiteness. The rigidity is checked by the Yukawa coupling.
Condition: Suppose that \(\pi :{\mathcal{X}}\to C\) is a family of polarized Calabi-Yau manifolds over a Riemann surface \(C\). Let \(t_0\in C\) such that for \(M=\pi^{-1}(t_0)\), the following condition is satisfied: for any non-zero \(\phi\in H^1(M,T^{1,0}(M))\), \(\wedge^n \phi\neq 0\) in \(H^n(M,(\Omega_M^n)^*)\). Then the above family is rigid. (Here \(n\) is the dimension of \(M\).)
In this paper, the moduli space of polarized Calabi-Yau manifolds and its Teichmüller space are studied.
Theorem 1. The Teichmüller space \(\tilde T(M)\) of a Calabi-Yau manifold \(M\) exists, and it has a finite number of components, each of which is a non-singular complex manifold. Furthermore, over \(\tilde T(M)\), there exists a universal family of marked polarized Calabi-Yau manifolds, up to the action of a finite group of automorphisms which acts trivially on the middle cohomology.
Theorem 2. There exists a finite cover \({\mathcal{M}}_L(M)\) of the coarse moduli space \({\mathfrak{M}}_L(M)\) of polarized Calabi-Yau manifolds such that \({\mathcal{M}}_L(M)\) is a non-singular quasi-projective variety and over \({\mathcal{M}}_L(M)\) there is a family of polarized Calabi-Yau manifolds.
Applications of these results are then discussed. Let \((C;\,x_1,\dots, x_n)\) be a Riemann surface with marked points \(x_1,\dots,x_n\). Let \(S\) be the divisor \(x_1+\cdots+x_n\) on \(C\). Let \(M\) be a Calabi-Yau manifold, fixed once and for all. Define \(\text{{Sh}}(C;S,M)\) to be the set of all possible families \({\mathcal{X}}\to C\) of Calabi-Yau manifolds defined up to an isomorphism over \(C\) with fixed degenerate fibers over the points \(x_1,\dots,x_n\), where \({\mathcal{X}}\) is a projective manifold and the generic fiber is a Calabi-Yau manifold \({\mathbf{C}}^{\infty}\) equivalent to \(M\).
Theorem 3. Suppose that the rigidity condition holds true for some fiber \(M_{\tau}\) of a family of Calabi-Yau manifolds over a fixed Riemann surface \(C\) with fixed points of degenerations. Then the set \(\text{{Sh}}(C;S)\) is finite.
The proof is in two steps: the first step is to show that \(\text{{Sh}}(C,S)\) is a discrete set, and the second step is to show that for each \(\phi\in\text{{Sh}}(C,S)\) the volume of \(\phi(C)\) is bounded by a universal constant. Then it follows that \(\text{{Sh}}(C,S)\) is compact. Since \(\text{{Sh}}(C,S)\) is discrete, the compactness implies that it is in fact a finite set.
Finally, counterexamples to the analogue of the Shafarevich conjecture are presented. Such an example is given by a non-rigid family of Calabi-Yau manifolds constructed from taking the product of non-isotrivial family of \(K3\) surfaces and a family of elliptic curves, and then passing to the quotients. polarized Calabi-Yau manifolds; Yukawa coupling; rigidity; Teichmüller space Liu, K; Todorov, A; Yau, S-T; etal., Shafarevich's conjecture for CY manifolds, Pure Appl Math Q, 1, 28-67, (2005) Calabi-Yau manifolds (algebro-geometric aspects), Algebraic moduli problems, moduli of vector bundles, Families, moduli, classification: algebraic theory, Calabi-Yau theory (complex-analytic aspects) Shafarevich's conjecture for CY manifolds I | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In the paper [Math. Ann. 347, No. 3, 689--702 (2010; Zbl 1193.13014)], the author and \textit{D. Ploog} found an explicit formula for Poincaré series of the algebra of invariants of a non $A_{2n}$ type finite subgroup of $\mathrm{SL}_2(\mathbb C)$ in terms of the characteristic polynomials of certain Coxeter elements determined by the corresponding (Kleinian) rational surface singularity. A similar formula was found for a Fuchsian singularity of genus 0 defined by the action of a Fuchsian group (of the first kind) on the tangent bundle of the upper half plane.
\par The paper under review studies some non-abelian finite subgroups $G \subset \mathrm{SL}_3(\mathbb C)$ defining non-isolated three-dimensional Gorenstein quotient singularities. The paper considers suitable hyperplane sections of such singularities which are Kleinian or Fuchsian surface singularities, and obtain a similar relation as above between the group $G$ and the corresponding surface singularity. McKay correspondence; Poincaré series; Kleinian singularities; Fuchsian singularities; Coxeter element Complex surface and hypersurface singularities, McKay correspondence, Actions of groups on commutative rings; invariant theory, Representation theory for linear algebraic groups A McKay correspondence for the Poincaré series of some finite subgroups of \(\mathrm{SL}_3(\mathbb C)\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We study the behaviour of representation varieties of quivers with relations under the operation of node splitting. We show how splitting a node gives a correspondence between certain closed subvarieties of representation varieties for different algebras, which preserves properties like normality or having rational singularities. Furthermore, we describe how the defining equations of such closed subvarieties change under the correspondence.
By working in the `relative setting' (splitting one node at a time), we demonstrate that there are many nonhereditary algebras whose irreducible components of representation varieties are all normal with rational singularities. We also obtain explicit generators of the prime defining ideals of these irreducible components. This class contains all radical square zero algebras, but also many others, as illustrated by examples throughout the paper. We also show that this is true when irreducible components are replaced by orbit closures, for a more restrictive class of algebras. Lastly, we provide applications to decompositions of moduli spaces of semistable representations of certain algebras. representations of algebras; quivers; determinantal varieties; determinantal ideals; rational singularities; node splitting; moduli spaces Representations of quivers and partially ordered sets, Linkage, complete intersections and determinantal ideals, Determinantal varieties, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Representation varieties of algebras with nodes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 noncommutative algebraic geometry, based on derived algebraic geometry, developed by Van den Bergh leads to the concept of Noncommutative Crepant Resolutions (NCCRs). This beautiful article brings this to its explicit form, and proves a very essential conjecture due to Bondal and Orlov in low dimensions:
If \(Y_1\) and \(Y_2\) are two crepant resolutions of \(X\), then \(Y_1\) is derived equivalent to \(Y_2\). In the study of one-dimensional fibres and in the McKay Correspondence for dimension \(d\leq 3\), \(Y_1\) and \(Y_2\) are derived equivalent to certain noncommutative rings. This means that to show \(Y_1\) derived equivalent to \(Y_2\) is equivalent to proving that the noncommutative rings are derived equivalent.
The authors really give all necessary definitions: If \(R\) is Cohen-Macaulay (CM), \(\Lambda\) a module-finite \(R\)-algebra, then {\parindent=6mm \begin{itemize} \item[(1)] \(\Lambda\) is called an \(R\)-order if \(\Lambda\) is maximal CM as \(R\)-module. It is called nonsingular if gl.dim \(\Lambda_{\mathfrak p}=\dim R_{\mathfrak p}\) for all primes \(\mathfrak p\subset R\). \item [(2)] A noncommutative Crepant Resolution (NCCR) of \(R\) is \(\Gamma=\text{End}_R(M)\) with \(M\) a non-zero reflexive \(R\)-module such that \(\Gamma\) is a non-singular \(R\)-order.
\end{itemize}} When \(R\) is CM and not necessarily Gorenstein, it can be many NCCR's of \(R\), and these are related to cluster tilting objects in the category CM \(R\).
This article considers a specialization of a more general conjecture: (Noncommutative Bondal-Orlov): If \(R\) is a normal Gorenstein domain, then all NCCRs of \(R\) are derived equivalent. This conjecture is generalized to some cases with CM rings, and the main result is (literally):
{Theorem}. Let \(R\) be a \(d\)-dimensional CM equi-codimensional normal domain with a canonical module \(\omega_R\). {\parindent=6mm \begin{itemize} \item[(1)] If \(d=2\), then all NCCRs of \(R\) are Morita equivalent. \item [(2)] If \(d=3\), then all NCCRs of \(R\) are derived equivalent.
\end{itemize}} An important point with this theorem, is that unlike other conditions in the literature, this condition is decided on the base singularity \(R\), and not on a fibre product.
Again, all necessary preliminaries are given, except possibly the theory of tilting modules. However, relevant references (equally good as the present article) to this are given. Even the definition of CM modules. Then the main result is explicitly proved by giving projective resolutions, and so the theorem follows directly from general results in tilting theory.
This article is the best practice to follow. noncommutative crepant resolutions; NCCR; orders; maximal Cohen Macaulay; non-singular order; derived equivalent M. Wemyss, \textit{Lectures on noncommutative resolutions}, arXiv:1210.2564 [INSPIRE]. Noncommutative algebraic geometry, Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.), Derived categories, triangulated categories, Global theory and resolution of singularities (algebro-geometric aspects) On the noncommutative Bondal-Orlov 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 A complex Lie group is called \textit{affine} if it admits a holomorphic homomorphism \(\alpha\colon G\to \mathrm{GL}(N,\mathbb{C})\), for some \(N\in \mathbb{Z}_{>0}\), such that the induced homomorphism between the respective Lie algebras is injective.
In this paper the authors consider Cartan geometries of type \((G,H)\), where \(G\) is a complex affine group and \(H\) is a complex Lie subgroup of \(G\), on a complex torus \(T\). They prove that
\textsl{Every holomorphic Cartan geometry of type \((G,H)\), on any complex torus is translation invariant.}
This results extends a similar result obtained by \textit{B. McKay} [SIGMA, Symmetry Integrability Geom. Methods Appl. 7, Paper 090, 11 p. (2011; Zbl 1247.53086)] for parabolic geometries (i.e. for pairs \((G,H)\), where \(G\) is a semisimple Lie group and \(H\) is a parabolic subgroup of \(G\)).
It was conjectured in [\textit{S. Dumitrescu} and \textit{B. McKay}, Complex Manifolds 3, 1--15 (2016; Zbl 1357.32017)], that \textsl{all flat Cartan geometries on complex tori are translation invariant}: there the conjecture was proved for tori of dimension \(\leq 2\) and for \(G\) nilpotent. complex torus; Cartan geometry Homogeneous complex manifolds, Homogeneous spaces and generalizations Holomorphic Cartan geometries on 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 The paper primarily accomplishes two tasks. On one hand, it serves as a review of the current status of `Mathieu Moonshine', and in particular the conjectures developed by \textit{T. Eguchi} et al. (EOT) [Exp. Math. 20, No. 1, 91--96 (2011; Zbl 1266.58008)] surrounding the Mathieu group's influence on the elliptic genus coefficients of \(K3\). On the other hand, the authors provide new results concerning the status of automorphism groups for cyclic torus orbifolds as subgroups of the Mathieu group, \(\mathbb{M}_{24}\). In particular, they show that the automorphism group of every cyclic torus orbifold \(K3\) sigma-model is not a subgroup of \(\mathbb{M}_{24}\), and also almost all \(K3\) sigma-models (with respect to possibilities of a classification theorem) whose automorphism group is not a subgroup of \(\mathbb{M}_{24}\) can be understood as cyclic torus orbifolds.
After the basics of Mathieu Moonshine are described, the authors explain how, in a sense, the conjectures of EOT are proved by a result of \textit{T. Gannon} [``Much ado about Mathieu'', \url{arXiv:1211.5531}] along with a (finite) number of explicit twisted genera constructions previously established by the authors and other researchers. As is pointed out, however, a deeper explanation, paralleling say the Monster vertex operator algebra in the `Monstrous Moonshine' phenomenon, has remained illusive.
To probe this question, a theorem previously developed by the two authors and \textit{S. Hohenegger} [Commun. Number Theory Phys. 6, No. 1, 1--50 (2012; Zbl 1275.81085)] is reviewed which classifies the groups of symmetries related to a non-linear sigma-model on \(K3\) which preserves the \(\mathcal{N}=(4,4)\) superconformal algebra and spectral flow operators. After sketching the proof of this theorem, the authors go on to show that (i) the group of symmetries of all \(K3\) sigma-models arising as orbifolds of a torus by a cyclic group are exceptional in the sense they are not subgroups of \(\mathbb{M}_{24}\), (ii) most models whose group of symmetries are exceptional can be found in this way, and (iii) there are, however, exceptional models that are not cyclic torus orbifolds.
Another large component of this paper is a construction of a \(\mathbb{Z}_5\) torus orbifold. Indeed, the steps of this construction are explicit and it is found that the corresponding full symmetry group is the semi-direct product \(5^{1+2}:\mathbb{Z}_4\), which is one of the cases of the aforementioned classification theorem and also exceptional. The authors conclude by explaining how models can be constructed whose group of symmetries correspond to all but one of the cases of the classification theorem. Mathieu Moonshine; orbifold; \(K3\) surfaces; elliptic genus Gaberdiel, MR; Volpato, R., \textit{Mathieu moonshine and orbifold K}3\(s\), Contrib. Math. Comput. Sci., 8, 109, (2014) \(K3\) surfaces and Enriques surfaces, Relationship to Lie algebras and finite simple groups, Simple groups: sporadic groups, Cohomology of groups Mathieu Moonshine and orbifold \(K3\)s | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This article can be regarded as a supplement to the following article \([D]\) by the same authors in Math. Ann. 288, No. 3, 527-547 (1990; Zbl 0715.32013).
Let \(X\) be a germ of a Cohen Macaulay surface singularity and \(Y\) be the image of \(X\) under a generically one-to-one map to \(\mathbb{C}^ 3\). The conductor ideal associated with the finite map \(X\to Y\) defines a subvariety \(\Sigma\) in \(Y\). In \([D]\) the relation between the deformation of the pair \(X\to Y\) and the deformation of the pair \(\Sigma\subset Y\) has been studied. In this article some applications of the results of \([D]\) to the theory of the so-called smoothable \(Q\)-Gorenstein singularities \((qG\)-singularities) are given. First an alternative verification of J. Steven's result ``A rational surface singularity of multiplicity four and index two is a \(qG\)-singularity'' is given. Second they consider the case where two different singularities \(X\) and \(X'\) have the same \(\Sigma\). (The corresponding \(Y\) and \(Y'\) may be different. But, \(\Sigma=\Sigma'\subset Y\cap Y'\subset\mathbb{C}^ 3\).) They show that in this case under some additional conditions there exists one-to-one correspondence between \(Q\)-Gorenstein smoothing components of \(X\) and those of \(X'\). Finally applying this result they construct several \(qG\)- singularities of index two starting from a simple non-normal singularity \(Y\) in \(\mathbb{C}^ 3\). Here \(Y\) is a union of several planes passing through the origin. surface singularity; \(Q\)-Gorenstein singularities; smoothing Deformations of complex singularities; vanishing cycles, Complex surface and hypersurface singularities, Local deformation theory, Artin approximation, etc. A construction of \(Q\)-Gorenstein smoothings of index 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 Let (X,x) be a germ of a normal complex surface singularity. In this note the author studies the set of isomorphism classes of vector bundles on the punctured spectrum \(X\setminus \{x\}\). These isomorphism classes are in a natural bijection with isomorphism classes of reflexive modules over the local ring \({\mathcal O}_{X,x}.\)
The first two chapters are a survey of the main facts about normal surface singularities, reflexive modules over the local ring of such a singularity and the Auslander-Reiten theory for modules.
The next chapters are the main technical kernel of this work. Let \(\pi\) : \(\tilde X\to X\) be a resolution of the singularity with exceptional fiber E and let Z be an effective divisor with support E. For any coherent reflexive sheaf M on X denote by \(R_ Z(M)=(\pi^*M)^{\vee}|_ Z\) (i.e. the restriction to Z of the bidual of \(\pi^*M)\). This is a locally free sheaf on Z. For a proper choice of the cycle Z (the so called reduction cycle), \(R_ Z\) defines an injective map from the set of isomorphism classes of reflexive \({\mathcal O}_{X,x}\)-modules to the set of isomorphism classes of locally free \({\mathcal O}_ Z\)-modules. The image of this map is characterized and explicitly described in the case of the rational singularities and minimal-elliptic singularities. In the case of the simply-elliptic singularites, when \(\pi\) is the minimal resolution, the exceptional curve E is a smooth irreducible elliptic curve and coincides with the reduction cycle. A complete characterization of the isomorphism classes of indecomposable reflexive modules over \({\mathcal O}_{X,x}\) is obtained using Atiyah's classification of the indecomposable vector bundles on an elliptic curve.
In chapter 6 the author proves that each vector bundle over the punctured spectrum of a simply-elliptic singularity is associated to a finite dimensional representation of the local fundamental group, which is a discrete Heisenberg group. A complete description of those representations which correspond to irreducible vector bundles is achieved.
The last chapter uses the above results for the study of the torsion free modules over plane curve singularities of type \(E_ 6\) and \(E_ 7\). Auslander-Reiten quiver; germ of a normal complex surface singularity; isomorphism classes of vector bundles; reflexive modules; reduction cycle; rational singularities; minimal-elliptic singularities; simply- elliptic singularites; torsion free modules over plane curve singularities Kahn, Reflexive Moduln auf einfach-elliptischen Flächensingularitäten, Dissertation (1988) Singularities of surfaces or higher-dimensional varieties, Singularities of curves, local rings, Representation theory of associative rings and algebras Reflexive Moduln auf einfach-elliptischen Flächensingularitäten. (Reflexive modules on simply elliptic surface singularities) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The memoir under review brings a crucial contribution to the theory of singular reduction. The natural framework of that theory is provided by the stratified symplectic spaces as shown e.g., in the paper by \textit{R.~Sjamaar} and \textit{E.~Lerman} [Ann. Math. (2) 134, No. 2, 375--422 (1991; Zbl 0759.58019)] and in the book by \textit{J.-P.~Ortega} and \textit{T. S.~Ratiu} [Momentum maps and Hamiltonian reduction (Progress in Mathematics 222, Birkhäuser, Boston, MA) (2004; Zbl 1241.53069)].
In the present paper the notion of stratified polarization is introduced and investigated. That notion plays a key role in order to understand the stratified Kähler spaces and provides the natural setting for the singular reduction of Kähler manifolds. More precisely, if a Kähler manifold is acted on by a compact Lie group such that the action is Hamiltonian, preserves the Kähler polarization and extends to an action of the complexified group, then the corresponding reduced space has a complex analytic stratified Kähler polarization. The latter structure also shows up in a quite different setting: the closure of any holomorphic nilpotent orbit in a semisimple Lie algebra of Hermitian type possesses a stratified Kähler polarization. The close relationship between these two sources of examples is thoroughly explained in section~5 of the paper. And the understanding of that relationship is then completed by exploring the connections with Jordan triple systems.
We would not conclude without mentioning that such a short review cannot do justice to the whole range of deep ideas contained in this excellent memoir. Poisson manifold; stratified space; holomorphic nilpotent orbit J. Huebschmann, Kähler spaces, nilpotent orbits, and singular reduction, Mem. Amer. Math. Soc., 172, (2004) Momentum maps; symplectic reduction, Geometric invariant theory, Group actions on varieties or schemes (quotients), Poisson algebras, Infinite-dimensional Lie (super)algebras, Associated manifolds of Jordan algebras Kähler spaces, nilpotent orbits, and singular reduction | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The aim of the paper is to generalize the well-known fact that any quasi-homogeneous normal complex surface singularity with \(\mathbb Q\)-homology sphere link has as the universal Abelian cover a Brieskorn complete intersection singularity [cf. \textit{W. D. Neumann}, Proc. Symp. Pure Math. 40, Part 2, 233--243 (1983; Zbl 0519.32010)].
More exactly, the authors prove a similar result for splice-quotient singularities, which are normal surface singularities with \(\mathbb Q\)-homology sphere links, and splice type singularities, which generalize Brieskorn complete intersections. The proof is based on detail analysis of systems of equations described in terms of splice diagrams, of the resolution dual graphs with their discriminant groups and some special congruence conditions.
In conclusion, the authors discuss their conjecture that rational singularities and minimally elliptic singularities with \(\mathbb Q\)-homology sphere links are splice-quotients and also remark that T. Okuma has recently proved it. normal surface singularities; \(\mathbb Q\)-Gorenstein singularities; rational homology sphere links; splice-quotients singularities; Brieskorn singularities; complete intersection singularities; Abelian cover Neumann W.D., Wahl J: Complete intersection singularities of splice type as universal abelian covers. Geom. Topol. 9, 699--755 (2005) Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants, Singularities in algebraic geometry, Knots and links in the 3-sphere, Topology of general 3-manifolds Complete intersection singularities of splice type as universal Abelian covers | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(d\ge 3\), \(n\ge 2\). The object of our study is the morphism \(\Phi \), introduced in earlier articles
by \textit{J. Alper} and the author [Math. Ann. 360, No. 3--4, 799--823 (2014; Zbl 1308.14048); J. Reine Angew. Math. 745, 83--104 (2018; Zbl 1403.14060)]; \textit{M. G. Eastwood} [Asian J. Math. 8, No. 2, 305--314 (2004; Zbl 1084.32019)] and \textit{M. G. Eastwood} and the author [Math. Ann. 356, No. 1, 73--98 (2013; Zbl 1277.32031)],
that assigns to every homogeneous form of degree \(d\) on \(\mathbb{C}^n\) for which the discriminant \(\Delta \) does not vanish a form of degree \(n(d-2)\) on the dual space, called the associated form. This morphism is \(\mathrm{SL}_n\)-equivariant and is of interest in connection with the well-known Mather-Yau theorem, specifically, with the problem of explicit reconstruction of an isolated hypersurface singularity from its Tjurina algebra. Letting \(p\) be the smallest integer such that the product \(\Delta^p\Phi \) extends to the entire affine space of degree \(d\) forms, one observes that the extended map defines a contravariant. In the present paper, we survey known results on the morphism \(\Phi \), as well as the contravariant \(\Delta^p\Phi \), and state several open problems. Our goal is to draw the attention of complex analysts and geometers to the concept of the associated form and the intriguing connection between complex singularity theory and invariant theory revealed through it. associated forms; isolated hypersurface singularities; Mather-Yau theorem; classical invariant theory; geometric invariant theory; contravariants of homogeneous forms Actions of groups on commutative rings; invariant theory, Geometric invariant theory, Complex surface and hypersurface singularities Associated forms: current progress and open problems | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(k\) be an algebraically closed field of characteristic zero. Orlov's representation theorem states that any fully faithful exact functor between the bounded derived categories of coherent sheaves on smooth projective varieties over \(k\) is a Fourier-Mukai functor. It is unknown if the hypothesis on fully faithfulness is necessary for this to be true. \textit{V. A. Lunts} and \textit{D. O. Orlov} [J. Am. Math. Soc. 23, No. 3, 853--908 (2010; Zbl 1197.14014)] extended the theorem to quasi-coherent sheaves: Let \(X/k\) be a projectve scheme s.t. \(\mathcal O_X\) has no zero-dimensional torsion, and let \(Y\) be a quasi-compact separated scheme. Then every fully faithful exact functor \(\Psi:\text{Perf}(X)\rightarrow D(\text{Qcoh}(Y))\) is isomorphic to the restriction of a Fourier-Mukai functor associated to an object in \(D(\text{Qcoh}(X\times Y))\).
A main result of this article is that the extended result above is false if the condition that \(\Psi\) is fully faithful is dropped. This is proved using scalar extensions of derived categories which are treated in the main body of the article.
If \(\mathfrak a\) is a \(k\)-linear category and \(B\) is a \(k\)-algebra, \(\mathfrak a_B\) denotes the category of \(B\)-objects in \(\mathfrak a\), i.e. pairs \((M,\rho)\) with \(M\in\text{Ob}(\mathfrak a)\), \(\rho:B\rightarrow\mathfrak a(M,M)\) a \(k\)-algebra homomorphism.
The first author studies the forgetful functor \(F:D^b(\mathcal C_B)\rightarrow D^b(\mathcal C)_B\) for a field extension \(B/k=L/k\). She proved a surjectivity result for \(\text{trdeg}L/k<2\).
The first result in the article is the following which is proved using \(A_\infty\)-techniques:
{Proposition B} Assume that \(\mathcal C\) is a Grothendieck category. If \(B/k\) has Hochschild dimension \(\leq 2\), then \(F\) is essentially surjective. If \(B/k\) has Hochschild dimension \(\leq 1\), \(F\) is (in addition) full. If \(B/k\) has Hochschild dimension \(0\), \(F\) is an equivalence of categories.
The next result proves that one cannot hope to substantially improve proposition B:
{Theorem C} Let \(X/k\) be a smooth connected projective variety which is not a point, a projective line or an elliptic curve. Then there exists a finitely generated field extension \(L/k\) of transcendence degree 3 together with an object \(Z\in D^b(\text{Qcoh(X))}_L\) which is not in the essential image of \(F\).
To prove this result, the authors prove a similar result for representations of wild quivers. Theorem C excludes the case where \(X\) is a curve of genus \(<1\). This is solved by introducing the \textit{essential dimension}, the minimal number of parameters required to define any family of indecomposable objects. It follows from this that if \(X\) is a curve of genus \(<1\) and \(\mathbb C=\text{Qcoh}(X)\), if \(L/k\) is a field extension, then the essential image of \(F\) contains all objects in \(D^b(\text{Qcoh}(X))_L\) whose objects is in \(\text{coh}(X_L)\subset\text{Qcoh}(X_L)\simeq\text{Qcoh}(X)_L\).
A counterexample to proposition A (dropping the faithfulness) is obtained by using the following:
{Theorem D}. Let \(X,Y\) be connected smooth projective schemes. Let \(i_\eta:\eta\rightarrow X\) be the generic point of \(X\), and let \(L=k(\eta)\) be the function field of \(X\). Assume that \(D^b(\text{Qcoh}(Y)_L)\) contains an object \(Z\) which is not in the essential image of \(D^b(\text{Qcoh}(Y)_L)\). Define \(\Psi\) as the composition \(\text{Perf}(X)\overset{i^\ast_\eta}\rightarrow D(L)\overset{L\mapsto Z}\rightarrow D(\text{Qcoh}(Y))\). Then \(\Psi\) is not the restriction of a Fourier-Mukai functor.
The main issue of the article is to prove the results above, and to give some guiding examples. The article starts by giving the necessary tools, including moduli spaces of representations of algebras, where the properties of the Formanek center and the concepts of Azumaya algebras as matrix algebras for the étale topology are involved, i.e. they are used to define a representable functor, thereby defining a moduli. This introduction to the theory is explicit and contains explicit results from algebra.
The algebraic theory is then lifted to vector bundles on curves, and the necessary results on stability is considered. For example, the following essential result of \textit{G. Faltings} [J. Algebr. Geom. 2, No. 3, 507--568 (1993; Zbl 0790.14019)] is used:
Let \(X\) be a smooth projective curve. A bundle \(\mathcal E\) on \(X\) is semi-stable if there exists a non-zero bundle \(\mathcal F\) such that \(\mathcal F\perp\mathcal E\).
Then the homological identities is given, using Hochshild cohomology. This gives a theory of lifting field actions in the hereditary case, and counterexamples to this. Then counterexamples to lifting in the geometric case can be given, as indicated in the introduction.
After that, \textit{Non-Fourier-Mukai} functors are defined, and some general results about them are given.
Finally, using all the introduced theory, the liftings can be studied using \(A_\infty\)-actions. This includes enhancing categories, studying \(A_\infty\)-schemes and \(A_\infty\)-modules by their \(A_\infty\)-morphisms. This not only proves the stated results, but gives very nice examples of the necessity of the \(A_\infty\)-theory.
A very nice and complete article, working in the recent edge of derived algebraic geometry. Fourier-Mukai functor; Orlov's theorem; A-infinity categories; Azumaya algebra; Formanek center; derived bounded categories Rizzardo, A.; Bergh, M., Scalar extensions of derived categories and non-Fourier-Mukai functors, Adv. Math., 281, 1100-1144, (2015) Derived categories and commutative rings, Derived categories, triangulated categories, Noncommutative algebraic geometry Scalar extensions of derived categories and non-Fourier-Mukai functors | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The purpose of this paper is to generalize a result about conic bundles to the case of \(\mathbb{P}^2\)-bundles. Let \(K\) be a field containing a primitive cubic root of unity \(\omega\), and \(A=(f,g)_{3,K}\) \((f,g\in K-\{0\})\) be a cyclic algebra of rank 9 over \(K\), i.e., \(A\) is generated by two elements \(x,y\) with relations \(x^3=f\), \(y^3=g\), \(yx= \omega xy\). Suppose there is an irreducible regular scheme \(X_0\) with the function field isomorphic to \(K\) satisfying the following condition (1). Denote the principal divisors of \(f\) and \(g\) by \((f)_{X_0} \equiv A\), \((g)_{X_0} \equiv B \text{mod} 3 \text{Div} (X_0)\), where \(A\) and \(B\) are sums of prime divisors of \(X_0\) with coefficients equal to one or two. Then
\[
(A+B)_{\text{red}} \text{ is a simple normal crossing divisor of } X_0. \tag{1}
\]
The main theorem of this paper is stated as follows.
Theorem. Under the above assumption (1), there exists a proper flat surjective morphism \(\tau: V\to X\) with \(V\) and \(X\) irreducible regular schemes such that
(i) \(X\) is obtained by blowing up \(X_0\),
(ii) the generic fibre of \(\tau\) is isomorphic to the \(K\)-form \(V_K\) of \(\mathbb{P}^2\) associated to the cyclic algebra \(A\),
(iii) the relative Picard group \(\text{Pic} (V/X)\) is isomorphic to \(\text{Pic} (V_K)\) by restriction.
If \(K\) is the function field of an algebraic variety over an algebraically closed field \(k\) of \(\text{char} (k)=0\), then there exists a smooth projective variety \(X_0\) over \(k\) satisfying (1) due to the theorem of resolution of singularities. Moreover, a central simple algebra of rank 9 over a field is always represented by a cyclic algebra by a theorem of Wedderburn. Hence we obtain as a corollary of our theorem above:
Corollary. Let \(K\) be the function field of an algebraic variety over an algebraically closed field \(k\) of \(\text{char} (k)=0\) and \(V_K\) be a \(K\)-form of \(\mathbb{P}^2\) (i.e., \(V_K \times_K \overline K\cong \mathbb{P}^2_K\) for an algebraic closure \(\overline K\) of \(K)\). Then there exists a proper flat surjective morphism \(\tau: V\to X\) with \(V\) and \(X\) smooth projective varieties such that
(i) the function field of \(X\) is isomorphic to \(K\),
(ii) the generic fibre of \(\tau\) is isomorphic to \(V_K\),
(iii) the relative Picard group \(\text{Pic} (V/X)\) is isomorphic to \(\text{Pic} (V_K)\) by restriction. projective plane bundles; principal divisors; normal crossing divisor; Picard group DOI: 10.1006/jabr.1997.7120 Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Divisors, linear systems, invertible sheaves, Projective techniques in algebraic geometry On standard projective plane bundles | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The main purpose of this paper is to relate the existence of a resolution of singularities to several classical problems in commutative algebra. The author first describes the uniform Artin-Rees theorem. The usual Artin-Rees lemma states that if \(R\) is Noetherian, \(N\subseteq M\) are finitely generated \(R\)-modules, and \(I\) is an ideal of \(R\) then there exists a \(k>0\) (depending on \(I,M,N)\) such that for all \(n>k\), \(I^nM\cap N=I^{n-k}(I^kM\cap N)\). A weaker statement which is often used is that for all \(n>k\), \(I^nM\cap N\subseteq I^{n-k}N\). When this containment holds for a fixed \(k\) and for all ideals \(I\), the pair \(N \subseteq M\) is said to satisfy the uniform Artin-Rees property. When all pairs \(N\subseteq M\) of finitely generated \(R\)-modules satisfy the uniform Artin-Rees property the ring \(R\) is said to have the uniform Artin-Rees property.
A main question is: Which rings have the uniform Artin-Rees property? The author made the following conjecture [\textit{C. Huneke}, Invent. Math. 107, No. 1, 203--223 (1992; Zbl 0756.13001)]:
Conjecture. Let \(R\) be an excellent Noetherian ring of finite Krull dimension. Let \(N\subseteq M\) be two finitely generated \(R\)-modules. Then there exists an integer \(k=k(N,M)\) such that for all ideals \(I \subseteq R\) and all \(n\geq k\),
\[
I^nM\cap N\subseteq I^{n-k}N.
\]
In this paper the author gives an affirmative answer to this question whenever \(R\) is a homomorphic image of a regular ring of finite Krull dimension and has the property that for all primes \(P\), the integral closure of \(R/P\) has a resolution of singularities obtained by blowing up an ideal. While it is true that the largest known class of rings satisfying this property are known to have the uniform Artin-Rees property by the main result of the paper cited above, the class of rings which have resolution of singularities is expected to include all excellent Noetherian domains. Moreover the method of proof gives considerable insight into related features of the uniform Artin-Rees problem. In particular the author finds generalizations of the following theorem of \textit{J. Lipman} and \textit{A. Sathaye} [Mich. Math. J. 28, 199--222 (1981; Zbl 0438.13019)]:
Let \(R\) be a regular local ring and let \(I\) be an ideal generated by \(\ell\) elements. Then for all \(n\geq\ell\), \(\overline{I^n}\subseteq I^{n-\ell+1}\), where \(\overline{I^n}\) is the integral closure of \(I^n\).
The author also gives some new results concerning the uniform annihilation of local cohomology. uniform Artin-Rees property; regular ring; finite Krull dimension; annihilation of local cohomology; Noetherian ring Huneke C. (2000). Desingularizations and the uniform Artin--Rees theorem. J. London Math. Soc. 62: 740--756 Structure, classification theorems for modules and ideals in commutative rings, Global theory and resolution of singularities (algebro-geometric aspects), Local cohomology and commutative rings, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Regular local rings Desingularizations and the uniform Artin-Rees theorem. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper studies sandwiched singularities in arbitrary characteristic. A normal 2-dimensional complete local ring \(\mathcal O\) (with algebraically closed residue field \(k\)) is \textit{sandwiched} if there exists a proper birational morphism \(X \to X_0\) from a normal algebraic surface \(X\) to a smooth surface \(X_0\) and a point \(x \in X\) such that \(\mathcal{O}\cong\widehat{\mathcal{O}_{X,x}}\). A normal point of an algebraic surface is a sandwiched singularity if its complete local ring is sandwiched. Over \(\mathbb C\) this definition is equivalent to that of \textit{M. Spivakovsky} [Ann. Math. (2) 131, No. 3, 411--491 (1990; Zbl 0719.14005)]. Given a point \(p\) on a smooth surface \(X_0\), let \(Y\to X_0\) be a sequence of point blowups centered above \(p\); contracting a connected divisor obtained by removing some components of the exceptional divisor gives a sandwiched singularity \((X, x)\) and any sandwiched singularity is locally étale isomorphic to such a singularity. The surface \(X\) is sandwiched between the smooth surfaces \(X_0\) and \(Y\), explaining the terminology.
The authors give six different characterisations of sandwiched singularities, in term of self-similarity properties. The last of them is the existence of a proper birational morphism of algebraic surfaces \(\pi\colon X'\to X\) such that \(\widehat{\mathcal{O}_{X',p}}\cong
\widehat{\mathcal{O}_{X,x}}\) for some point \(p\in \pi^{-1}(x)\). This property can also be formulated in terms of the dual graph of the
singularity. A connected weighted graph is self-similar if it admits a nontrivial modification (where an elementary modification corresponds to a point blowup) containing a subgraph isomorphic to the original graph.
The other characterisations involve the non-archimedean link \(\operatorname{NL}(X, x)\) of \((X, x)\). If \(k\) is endowed with the
trivial absolute value and \(X^{\text{an}}\) is the analytification in the sense of Berkovich geometry, then \(\operatorname{NL}^\varepsilon(X, x) = \{y\in X^{\text{an}}\mid \max_i |z_i (y)| = \varepsilon\}\) does not depend on the embedding nor on \( \varepsilon\in (0,1)\) and is called
the non-archimedean link of \((X, 0)\). Concretely it is is the set of semi-valuations \(v\) on \(\widehat{\mathcal{O}_{X,x}}\) that are
normalized by the condition \(\min_{f\in\mathfrak{m}}v( f ) = 1\). A detailed description of the analytic structure of \(\operatorname{NL}(X, x)\) is given, following the work of the first author [Trans. Am. Math. Soc. 370, No. 11, 7815--7859 (2018; Zbl 1423.14171)]. Self-similarity of the non-archimedean link \(\operatorname{NL}(X, x)\), and by extension of \((X, x)\), means that there exists a finite set \(T\) of points of
type \(1\) (endpoints corresponding to semi-valuations with nontrivial kernel) of \(\operatorname{NL}(X, x)\) such that \(\operatorname{NL}(X, x)\setminus T\) is isomorphic to an open subset \(U\) of \(\operatorname{NL}(X, x)\) with \(\overline U\) a proper subset of \(\operatorname{NL}(X, x)\). This condition is implied by the existence of a finite set \(T\) of points of type \(1\) such that every open
subset of \(\operatorname{NL}(X, x)\) contains an open subset isomorphic to \(\operatorname{NL}(X, x)\setminus T\).
The authors also searched for a characterization of sandwiched singularities in terms of their archimedean links. A self-similar property reminiscent of the last mentioned property was not found. Building on the first self-similarity property the authors prove that a (complex analytic) normal surface singularity is sandwiched if and only if its (archimedean) link can be realised as a real-analytic global strongly pseudoconvex 3-fold in a compact (non-Kähler) complex surface containing a global spherical shell. Here global means that the complement of the real 3-fold or of the spherical shell is connected. non-archimedean link; sandwiched singularity; self-similarity Singularities in algebraic geometry, Non-Archimedean analysis, Local complex singularities Links of sandwiched surface singularities and self-similarity | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 originates in part from the Habilitation mémoire (Université de Bourgogne, Dijon, 2016) of the author and in part from an introductory talk he gave at the RAQIS'16 conference held at Geneva, Switzerland, in August 2016. It deals with a novel construction that associates an integrable, tau-symmetric hierarchy and its quantization to a cohomological field theory on the moduli space of stable curves, without the semi-simplicity assumption which is needed for the Dubrovin-Zhang hierarchy. It is inspired by Eliashberg, Givental and Hofer's symplectic field theory [\textit{Y. Eliashberg} et al., in: GAFA 2000. Visions in mathematics---Towards 2000. Proceedings of a meeting, Tel Aviv, Israel, August 25--September 3, 1999. Part II. Basel: Birkhäuser. 560--673 (2000; Zbl 0989.81114)] and is the fruit of a joint project of the author with A. Buryak, B. Dubrovin and J. Guéré (see e.g. [\textit{A. Buryak}, et al. ``Tau-structure for the double ramification hierarchies'', \url{arXiv:1602.05423}; ``Integrable systems of double ramification type'', \url{arXiv:1609.04059}]).
After a self contained introduction to the language of integrable systems in the formal loop space and the needed notions from the geometry of the moduli space of stable curves the author explains the double ramification hierarchy construction and presents its main features, with an accent on the quantization procedure, concluding with a list of examples worked out in detail. This paper does not contain new results and most of this material originates from an ongoing collaboration with A. Buryak, B. Dubrovin and J. Guéré. It is however a complete reorganization and, in part, a rephrasing of those results with the aim of showcasing the power of these methods and making them more accessible to the mathematical physics community. moduli space of stable curves; integrable systems; cohomological field theories; double ramification cycle; double ramification hierarchy Rossi, P., Integrability, quantization and moduli spaces of curves, SIGMA, 13, (2017) Families, moduli of curves (algebraic), Relationships between algebraic curves and integrable systems, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) Integrability, quantization and moduli spaces of curves | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper deals with simple algebraic groups \(G\) over \(\mathbb{C}\) and their faithful irreducible rational finite-dimensional representations \(\varphi\colon G\to\text{GL}(V)\). A set of algebraic equations defining \(G\) as a subvariety of \(\text{GL}(V)\) is derived. Consider the following table of Dynkin schemes for \(\varphi\):
\[
\circ\diagrbar\circ\rightrightarrows\overset {1}\circ\qquad\circ\diagrbar\circ\cdots\circ\diagrbar\circ\rightrightarrows\overset {k}\circ\;k\text{ is even, rank is odd, rank}>2\qquad\circ\Rrightarrow\overset {k}\circ.
\]
Let \(X\subset\mathbb{P}(V)\) be the \(G\)-orbit of the highest weight line \([v]\) in projective space \(\mathbb{P}(V)\). The author derives that if \(\varphi\colon G\to\text{GL}(V)\) is other than those in the table, then \(G\) is the set of all linear transformations \(\tau\in\text{GL}(V)\) satisfying the following equations:
(i) \(\text{det }\tau=1\); and in the case \(\varphi\) is selfdual, \((\tau v_1,\tau v_2)=(v_1,v_2)\) for any \(v_1,v_2\in V\). Here \((\;,\;)\) is a \(G\)-invariant bilinear form on \(V\).
(ii) \(p(\tau w)=0\) for the generalized Plücker polynomials \(p\) on \(V\) and for \(w\in Gv\). Here \(Gv\) is the affine cone over \(X\).
If \(\varphi\colon G\to\text{GL}(V)\) is one of those from the table, then the set of equations is obtained, too. simple algebraic groups; faithful irreducible rational finite-dimensional representations Representation theory for linear algebraic groups, Classical groups (algebro-geometric aspects), Lie algebras of linear algebraic groups, Affine algebraic groups, hyperalgebra constructions On the equations defining irreducible algebraic groups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities For every genus \(g\geq4\), the dimension \(3g-3\) of the moduli space \(\mathcal{J}_{g}\) of Riemann surfaces (or equivalently Jacobians, by Torelli's Theorem) is strictly smaller that the dimension \(\frac{g(g+1)}{2}\) of the moduli space \(\mathcal{A}_{g}\) of principally polarized abelian varieties. A natural question is then to find, in various subsets of \(\mathcal{A}_{g}\) having algebro-geometric meaning, equations describing their intersection of \(\mathcal{A}_{g}\). This problem is named after Schottky, who solved it for genus 4 (where the co-dimension of \(\mathcal{J}_{g}\) in \(\mathcal{A}_{g}\) is 1). The resulting equation, as well as the (full or partial) solutions to Schottky type problems, involves theta functions, as such a function is the unique gloabl section (up to scalars) of the line bundle associated with the principal polarization.
For each principally polarized abelian variety \(B\), the associated theta function defines a theta divisor on \(B\), which is singular if the point in \(\mathcal{A}_{g}\) that is associated with \(B\) lies in a particular divisor on \(\mathcal{A}_{g}\). This divisor consists of the irreducible divisor \(\theta_{\mathrm{null}}\) parameterizing principally polarized abelian varieties whose theta divisor contains a point of order 2 (thus a vanishing even theta constant) together with another irreducible divisor, and when \(g\geq4\) if contains \(\mathcal{J}_{g}\). In addition, a singular point \(P\) determines an invariant by considering the rank of the Hessian matrix of the theta function at \(P\), and one defines the closed subset \(\theta_{\mathrm{null}}^{h}\) consisting of those points in \(\theta_{\mathrm{null}}\) in which the Hessian matrix has rank \(\leq h\). It is known that \(\mathcal{J}_{g}\cap\theta_{\mathrm{null}}\) is contained in \(\theta_{\mathrm{null}}^{3}\), with equality holding when \(g=4\). The main result of this paper is that this intersection is an irreducible component of the set \(\theta_{\mathrm{null}}^{3}\).
Recall that the Baily-Borel compactification of \(\mathcal{A}_{g}\) is obtained by adding, for each \(l \leq g\), a unique cusp that is isomorphic to \(\mathcal{A}_{l}\). Moreover, any other compactification of \(\mathcal{A}_{g}\) (in particular toroidal ones) admits a unique map to the Baily-Borel compactification, and for a toroidal compactification, the inverse image of the largest cusp \(\mathcal{A}_{g-1}\) is canonical, i.e., independent of choices that are typically required for defining toroidal compactifications. It is always isomorphic to the universal family \(\mathcal{X}_{g-1}\) over \(\mathcal{A}_{g-1}\), in which the fiber over a generic point is the Kummer variety associated with the abelian variety of dimension \(g-1\) that it parameterizes (because \(-\operatorname{Id}\) is in \(\operatorname{Sp}_{2g}(\mathbb{Z})\)). The proof of the main theorem is based on working in the partial canonical compactification \(\mathcal{A}_{g}\cup\mathcal{X}_{g-1}\), and bounding the dimension of the intersection of \(\theta_{\mathrm{null}}^{3}\) with the boundary part.
The details of this analysis involve the cover \(\mathcal{A}_{g}(2)\) obtained from the congruence subgroup of level 2 in \(\operatorname{Sp}_{2g}(\mathbb{Z})\), the Gauss map from \(\mathcal{X}_{g-1}\) to the corresponding projective tangent space, and the resulting Thom-Boardman loci, all extended to the canonical toroidal boundary component. These are applied for obtaining a lower bound for the co-dimension of an irreducible component \(\mathcal{Z}_{g}\) of \(\theta_{\mathrm{null}}^{3}\) that contains \(\mathcal{J}_{g}\cap\theta_{\mathrm{null}}\), by investigating its intersection with the universal double theta divisor \(X_{g}\) in \(\mathcal{X}_{g-1}\subseteq\mathcal{A}_{g}\cup\mathcal{X}_{g-1}\). Some of the first steps can be done for any \(g\geq4\), but the later steps work only for \(g=5\) (indeed, the authors show that if \(g\geq6\) then a certain subset of \(\mathcal{A}_{g-1}\) is no longer strictly contained in \(\mathcal{A}_{g-1}\), a condition that is crucial for the proof). The authors conclude with an example, evaluated using the Julia package for theta functions.
The paper is divided into 5 sections. Section 1 is the Introduction, with the required definitions and the statement of the main theorem. Section 2 includes the notation and the definition of the basic objects. Section 3 considers the details of the divisor \(\theta_{\mathrm{null}}\) on \(\mathcal{A}_{g}\), as well as on its inverse image in \(\mathcal{A}_{g}(2)\). Section 4 presents the partial compactification, as well as and the Thom-Boardman loci of the Gauss map. Finally, Section 5 proves the main theorem, and gives the example. theta functions; abelian varieties; Jacobians; Schottky problem Theta functions and curves; Schottky problem, Theta functions and abelian varieties, Algebraic moduli of abelian varieties, classification On the Schottky problem for genus five Jacobians with a vanishing theta null | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 two-dimensional complex case a log-terminal singularity is a singularity of the form \({\mathbb{C}}^ 2/G\), where G is a finite subgroup of GL(2,\({\mathbb{C}})\). A normal projective surface Z over \({\mathbb{C}}\) is called del Pezzo surface with log-terminal singularities, or for short log-del Pezzo surface, if all singularities of Z are log-terminal and for some positive integer k the multiple \(-kK_ Z\) of the anticanonical Weil divisor \(-K_ Z\) of Z is an ample Cartier divisor. The smallest such number k is called the index of the log-del Pezzo surface Z. For a minimal desingularization \(\sigma:\quad Y\to Z\) of Z we have \(K_ Y=\sigma^*K_ Z+\sum_{i}\alpha_ iF_ i \) where \(F_ i\) are irreducible components of the exceptional divisor of \(\sigma\) (all of them are non-singular and rational) and \(\alpha_ i\in {\mathbb{Q}}\), \(- 1<\alpha_ i\leq 0\). Then the smallest common denominator of the numbers \(\alpha_ i\) is the index k. One shows that Y is a rational surface. A complete classification of log-del Pezzo surfaces, which lies in giving the list of all possible intersection diagrams of the curves \(F_ i\), is known in the log-singular case and in case of indices \(k=1\quad and\quad 2\) only.
The author proves some facts about various invariants of log-del Pezzo surfaces and their minimal desingularizations such as the intersection diagrams, the rank \(\rho\) (Y) of the Picard lattice of Y, the multiplicities of the singular points of Z, the numbers \(\alpha_ i\) and the \(index\quad k.\) Here is his main result: Let for every singular point of Z of an index \(>1\) the corresponding \(\alpha_ i\) be smaller than - 1/2. Denote by e the maximal multiplicity of all singular points of Z. Then \(\rho (Y)\leq 352e+1284\leq 704k+1284.\)
The author also gives a conjecture which, as he writes in a note added in proof, has been proved in another paper: For a fixed k, \(\rho\) (Y) is bounded from above by a constant depending on k only, in particular the number of the intersection diagrams is then finite. Apart from this the author gives other conjectures and many results which are too technical, however, to be described in a review. Lobachevski space; del Pezzo surface with log-terminal singularities; anticanonical Weil divisor; log-del Pezzo surfaces; minimal desingularizations; intersection diagrams; Picard lattice Nikulin, V.V.: Del Pezzo surfaces with log-terminal singularities. Mat. Sb. 180(2), 226--243 (1989) [translation in Math. USSR-Sb. 66(1), 231--248 (1990)] Singularities of surfaces or higher-dimensional varieties, \(K3\) surfaces and Enriques surfaces, Transcendental methods of algebraic geometry (complex-analytic aspects), Picard groups, Moduli, classification: analytic theory; relations with modular forms, Reflection groups, reflection geometries, Global theory and resolution of singularities (algebro-geometric aspects) Del Pezzo surfaces with log-terminal singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author provides interesting, useful calculations of a part of the group homology, with \(\mathbb{Z}[1/2]\) coefficients, of \(\mathrm{SL}_2(k[C])\) where \(C\) is an affine curve over the algebraically closed field \(k\). Let \(\bar{C}\) be a smooth projective curve over \(k\), let \(P_1,\dots, P_s\) be closed points of \(\bar{C}\) and let \(C\) be the affine curve \(\bar{C}\setminus\{ P_1,\dots, P_s\}\). Let \(\Gamma = \mathrm{SL}_2(k[C])\) and let \(\mathfrak{X}_C\) be the associated building. \(\mathfrak{X}_C\) is a product of trees \(\mathfrak{T}_1\times \cdots \times \mathfrak{T}_s\) where \(\mathfrak{T}_i\) is the tree of the valuation \(v_i\) associated to \(P_i\). The vertices of the orbit space \(\Gamma\backslash\mathfrak{X}_C\) are naturally indexed by certain equivalence classes of rank \(2\) vector bundles \(\mathcal{E}\) on \(\bar{C}\) whose restriction to \(C\) is trivial. The author introduces the \textit{parabolic subcomplex} \(\mathfrak{P}_C\) of \(\mathfrak{X}_C\) consisting of those cells whose stabilizer in \(\Gamma\) contains a non-central non-unipotent element. The orbits of the vertices of \(\mathfrak{P}_C\) are then precisely the equivalence classes of rank \(2\) bundles which decompose as a sum of line bundles. The quotient of \(\mathfrak{X}_C\) modulo the subcomplex \(\mathfrak{P}_C\) is denoted \(\mathfrak{U}_C\), the `unknown quotient'. There is thus a long exact sequence in equivariant homology \(\dots \to H_\bullet^{\Gamma}(\mathfrak{P}_C)\to H_\bullet(\Gamma)\to H_\bullet^{\Gamma}(\mathfrak{U}_C)\to \cdots\) (the middle term arising from the contractibility of \(\mathfrak{X}_C\)). The main result of the paper is then a computation of the equivariant homology groups \(H_\bullet^{\Gamma}(\mathfrak{P}_C, \mathbb{Z}[1/2])\) of the parabolic complex, obtained by carefully analysing the structure of the orbit space \(\Gamma\backslash \mathfrak{P}_C\) and the terms of the related isotropy spectral sequence: Let \(\iota\) denote the involution \([L]\mapsto [L^{-1}]\) of the Picard group \(\mathrm{Pic}(C)\). Then \(H_\bullet^{\Gamma}(\mathfrak{P}_C, \mathbb{Z}[1/2])\cong \bigoplus_{[L]\in \mathrm{Pic}(C)/\iota }H_\bullet^{\Gamma}(\mathfrak{P}_C(L),\mathbb{Z}[1/2])\). Here \(\mathfrak{P}_C(L)\) is the connected component of \(\mathfrak{P}_C\) corresponding to \([L]\) and the groups \(H_\bullet^{\Gamma}(\mathfrak{P}_C(L), \mathbb{Z}[1/2])\) can be described in an accessible way: If \(L|_C\not\cong L^{-1}|_C\) then \(H_\bullet^{\Gamma}(\mathfrak{P}_C(L), \mathbb{Z}[1/2])\cong H_\bullet(k[C]^\times, \mathbb{Z}[1/2])\), while if \(L|_C\cong L^{-1}|_C\) there is a long exact sequence of the form \(\dots \to H_\bullet(SN,\mathbb{Z}[1/2])\to H_\bullet^{\Gamma}(\mathfrak{P}_C(L), \mathbb{Z}[1/2])\to \mathcal{RP}^1_\bullet(k)\otimes_\mathbb{Z}\mathbb{Z}[1/2][k[C]^\times/(k[C]^\times)^2]\to \cdots\) where \(SN\) denotes the group of monomial matrices in \(\Gamma\) and the groups \(\mathcal{RP}^1_\bullet(k)\) are `refined scissors congruence groups' defined by the author. These latter groups can be understood quite explicitly in low dimensions.
The author treats also the case \(\Gamma=\mathrm{GL}_2(k[C])\). Furthermore, he shows how to extend the results in some special cases -- eg \(C=\mathbb{P}^1(k)\setminus\{ 0,\infty\}\) -- to arbitrary infinite fields \(k\). The author also discusses the relationship of his computations, upon taking limits, to the predictions of the Friedlander-Milnor conjecture for \(\mathrm{SL}_2(k(C))\). group homology; special linear group; rank one M. Wendt, Homology of \(\operatorname{SL}_2\) over function fields II: rational function fields, in preparation. Cohomology theory for linear algebraic groups, Linear algebraic groups over arbitrary fields, Groups acting on trees, Group actions on varieties or schemes (quotients), Equivariant homology and cohomology in algebraic topology, Homology and cohomology of Lie groups Homology of \(\operatorname{SL}_2\) over function fields. I: Parabolic subcomplexes | 0 |
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