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Register a new userCrepant resolutions and Hilb^G(C^4) for certain abelian subgroups for SL(4,C)
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Y.Sato
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Let G be a finite subgroup of SL(n,C), then the quotient C^n/G has a Gorenstein canonical singularity. Bridgeland-King-Reid proved that the G-Hilbert scheme Hilb^G(C^3) gives a crepant resolution of the quotient C^3/G for any finite subgroup G of SL(3,C). However, in dimension 4, very few crepant resolutions are known. In this paper, we will show several examples of crepant resolutions in dimension 4 and show examples in which Hilb^G(C^4) is blow-up of certain crepant resolutions for C^4/G, or Hilb^G(C^4) has singularity.
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For any finite subgroup G in SL3(C), work of Bridgeland-King-Reid constructs an equivalence between the G-equivariant derived category of C^3 and the derived category of the crepant resolution Y = G-Hilb(C^3) of C^3/G. When G is abelian we show that this equivalence gives a natural correspondence between irreducible representations of G and certain sheaves on exceptional subvarieties of Y, thereby extending the McKay correspondence from two to three dimensions. This categorifies Reids recipe and extends earlier work from [CL09] and [Log10] which dealt only with the case when C^3/G has one isolated singularity.
The $q$-analog of Kostants weight multiplicity formula is an alternating sum over a finite group, known as the Weyl group, whose terms involve the $q$-analog of Kostants partition function. This formula, when evaluated at $q=1$, gives the multiplicity of a weight in a highest weight representation of a simple Lie algebra. In this paper, we consider the Lie algebra $mathfrak{sl}_4(mathbb{C})$ and give closed formulas for the $q$-analog of Kostants weight multiplicity. This formula depends on the following two sets of results. First, we present closed formulas for the $q$-analog of Kostants partition function by counting restricted colored integer partitions. These formulas, when evaluated at $q=1$, recover results of De Loera and Sturmfels. Second, we describe and enumerate the Weyl alternation sets, which consist of the elements of the Weyl group that contribute nontrivially to Kostants weight multiplicity formula. From this, we introduce Weyl alternation diagrams on the root lattice of $mathfrak{sl}_4(mathbb{C})$, which are associated to the Weyl alternation sets. This work answers a question posed in 2019 by Harris, Loving, Ramirez, Rennie, Rojas Kirby, Torres Davila, and Ulysse.
We define the branched analog of SL(r,C)-opers and investigate their properties. For the usual SL(r,C)-opers, the underlying holomorphic vector bundle is independent of the opers. For the branched SL(r,C)-opers, the underlying holomorphic vector bundle depends on the oper. Given a branched SL(r,C)-oper, we associate to it another holomorphic vector bundle equipped with a logarithmic connection. This holomorphic vector bundle does not depend on the branched oper. We characterize the branched SL(r,C)-opers in terms of the logarithmic connections on this fixed holomorphic vector bundle.
We say that an exact equivalence between the derived categories of two algebraic varieties is tilting-type if it is constructed by using tilting bundles. The aim of this article is to understand the behavior of tilting-type equivalences for crepant resolutions under deformations. As an application of the method that we establish in this article, we study the derived equivalence for stratified Mukai flops and stratified Atiyah flops in terms of tilting bundles.
Folding of ADE-Dynkin diagrams according to graph automorphisms yields irreducible Dynkin diagrams of ABCDEFG-types. This folding procedure allows to trace back the properties of the corresponding simple Lie algebras or groups to those of ADE-type. In this article, we implement the techniques of folding by graph automorphisms for Hitchin integrable systems. We show that the fixed point loci of these automorphisms are isomorphic as algebraic integrable systems to the Hitchin systems of the folded groups away from singular fibers. The latter Hitchin systems are isomorphic to the intermediate Jacobian fibrations of Calabi--Yau orbifold stacks constructed by the first author. We construct simultaneous crepant resolutions of the associated singular quasi-projective Calabi--Yau threefolds and compare the resulting intermediate Jacobian fibrations to the corresponding Hitchin systems.
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