Crepant Resolution Research Articles

The orbifold quantum cohomology of $\mathbb{C}^{2}/\mathbb{Z}_3$ and Hurwitz-Hodge integrals

Let Z_3 act on C^2 by non-trivial opposite characters. Let X =[C^2/Z_3] be the orbifold quotient, and let Y be the unique crepant resolution. We show the equivariant genus 0 Gromov-Witten potentials of X and Y are equal after a change of variables -- verifying the Crepant Resolution Conjecture for the pair (X,Y). Our computations involve Hodge integrals on trigonal Hurwitz spaces which are of independent interest. In a self contained Appendix, we derive closed formulas for these Hurwitz-Hodge integrals.

Endomorphism Rings of Finite Global Dimension

AbstractFor a commutative local ring R, consider (noncommutative) R-algebras Λ of the form Λ = EndR(M) where M is a reflexive R-module with nonzero free direct summand. Such algebras Λ of finite global dimension can be viewed as potential substitutes for, or analogues of, a resolution of singularities of Spec R. For example, Van den Bergh has shown that a three-dimensional Gorenstein normal ℂ-algebra with isolated terminal singularities has a crepant resolution of singularities if and only if it has such an algebra Λ with finite global dimension and which is maximal Cohen–Macaulay over R (a “noncommutative crepant resolution of singularities”). We produce algebras Λ = EndR(M) having finite global dimension in two contexts: when R is a reduced one-dimensional complete local ring, or when R is a Cohen–Macaulay local ring of finite Cohen–Macaulay type. If in the latter case R is Gorenstein, then the construction gives a noncommutative crepant resolution of singularities in the sense of Van den Bergh.

NONEXISTENCE OF A CREPANT RESOLUTION OF SOME MODULI SPACES OF SHEAVES ON A K3 SURFACE

Let $M_c=M(2,0,c)$ be the moduli space of O(1)-semistable rank 2 torsion-free sheaves with Chern classes $c_1=0$ and $c_2=c$ on a K3 surface $X$ where O(1) is a generic ample line bundle on $X$. When $c=2n\geq4$ is even, $M_c$ is a singular projective variety equipped with a symplectic structure on the smooth locus. In this paper, we show that there is no crepant resolution of $M_{2n}$ for $n\geq 3$. This implies that there is no symplectic desingularization.

Smooth toric G-Hilbert schemes via G-graphs

We provide here an infinite family of finite subgroups { G n ⊂ SL n ( C ) } n ⩾ 2 for which the G-Hilbert scheme G n - Hilb A n is a crepant resolution of A n / G n , via the Hilbert–Chow morphism. The proof is based on an explicit description of the toric structure of G n - Hilb A n in terms of Nakamura's G n -graphs. To cite this article: M. Sebestean, C. R. Acad. Sci. Paris, Ser. I 344 (2007).

Fractional two-branes, toric orbifolds and the quantum McKay correspondence

We systematically study and obtain the large-volume analogues of fractional two-branes on resolutions of orbifolds C^3/Z_n. We study a generalisation of the McKay correspondence proposed in hep-th/0504164 called the quantum McKay correspondence by constructing duals to the fractional two-branes. Details are explicitly worked out for two examples -- the crepant resolutions of C^3/Z_3 and C^3/Z_5.

Auslander correspondence

We study Auslander correspondence from the viewpoint of higher-dimensional analogue of Auslander–Reiten theory [O. Iyama, Higher dimensional Auslander–Reiten theory on maximal orthogonal subcategories, Adv. Math. 210 (1) (2007) 22–50 (this issue)] on maximal orthogonal subcategories. We give homological characterizations of higher dimensional analogue of Auslander algebras in terms of global dimension, Auslander-type conditions and so on. Especially we give an answer to a question of M. Artin [M. Artin, Maximal orders of global dimension and Krull dimension two, Invent. Math. 84 (1) (1986) 195–222]. They are also closely related to Auslander's representation dimension of Artin algebras [M. Auslander, Representation dimension of Artin algebras, in: Lecture Notes, Queen Mary College, London, 1971] and Van den Bergh's non-commutative crepant resolutions of Gorenstein singularities [M. Van den Bergh, Non-commutative crepant resolutions, in: The Legacy of Niels Henrik Abel, Springer, Berlin, 2004, pp. 749–770].

Brauer groups and crepant resolutions

We suggest a twisted version of the categorical McKay correspondence and prove several results related to it.

Hodge structures for orbifold cohomology

We construct a polarized Hodge structure on the primitive part of Chen and Ruan’s orbifold cohomology H o r b k ( X ) H_{orb}^k(X) for projective S L SL -orbifolds X X satisfying a “Hard Lefschetz Condition”. Furthermore, the total cohomology ⨁ p , q H o r b p , q ( X ) \bigoplus _{p,q}H_{orb}^{p,q}(X) forms a mixed Hodge structure that is polarized by every element of the Kähler cone of X X . Using results of Cattani-Kaplan-Schmid this implies the existence of an abstract polarized variation of Hodge structure on the complexified Kähler cone of X X . This construction should be considered as the analogue of the abstract polarized variation of Hodge structure that can be attached to the singular cohomology of a crepant resolution of X X , in light of the conjectural correspondence between the (quantum) orbifold cohomology and the (quantum) cohomology of a crepant resolution.

DESINGULARIZATIONS OF CALABI–YAU 3-FOLDS WITH A CONICAL SINGULARITY

We study Calabi-Yau 3-folds M_0 with a conical singularity x modelled on a Calabi-Yau cone V. We construct desingularizations of M_0, obtaining a 1-parameter family of compact, nonsingular Calabi-Yau 3-folds which has M_0 as the limit. The way we do is to choose an Asymptotically Conical Calabi-Yau 3-fold Y modelled on the same cone V, and then glue into M_0 at x after applying a homothety to Y. We then get a 1-parameter family of nearly Calabi-Yau 3-folds M_t depending on a small real variable t. For sufficiently small t, we show that the nearly Calabi-Yau structures on M_t can be deformed to genuine Calabi-Yau structures, and therefore obtaining the desingularizations of M_0. Our result can be applied to resolving orbifold singularities and hence provides a quantitative description of the Calabi-Yau metrics on the crepant resolutions.

Résolutions de singularités non-abéliennes en dimension trois

Let E be an elliptic curve and R be a real three dimensional root system. Let W be the Weyl group associated to R. Denote W + = W ∩ SL 3 ( C ) . The quotient E ⊗ Q ( R ) / W + admits two natural crepant resolutions. One is the result of a Jung process of desingularization of singularities, the other the equivariant Hilbert scheme. To compare these resolutions, we calculate the fibres over any singular point in both cases. We exhibit a McKay correspondence phenomenon for the Hilbert scheme resolution and construct a new family of vector bundles on E parametrized by the W + -Hilbert scheme. To cite this article: S. Térouanne, C. R. Acad. Sci. Paris, Ser. I 342 (2006).

On the existence of a crepant resolution of some moduli spaces of sheaves on an abelian surface

Let J be an abelian surface with a generic ample line bundle Open image in new window. For n≥1, the moduli space MJ(2,0,2n) of Open image in new window(1)-semistable sheaves F of rank 2 with Chern classes c1(F)=0, c2(F)=2n is a singular projective variety, endowed with a holomorphic symplectic structure on the smooth locus. In this paper, we show that there does not exist a crepant resolution of MJ(2,0,2n) for n≥2. This certainly implies that there is no symplectic desingularization of MJ(2,0,2n) for n≥2.

Towards the multiplicative behavior of the K-theoretical McKay correspondence

When the quotient of a symplectic vector space by the action of a finite subgroup of symplectic automorphisms admits as a crepant projective resolution of singularities the Hilbert scheme of regular orbits of Nakamura, then there is a natural isomorphism between the Grothendieck group of this resolution and the representation ring of the group, given by the Bridgeland-King-Reid map. However, this isomorphism is not compatible with the ring structures. For the Hilbert scheme of points on the affine plane, we study the multiplicative behavior of this map.

The orbifold Chow ring of toric Deligne-Mumford stacks

Generalizing toric varieties, we introduce toric Deligne-Mumford stacks. The main result in this paper is an explicit calculation of the orbifold Chow ring of a toric Deligne-Mumford stack. As an application, we prove that the orbifold Chow ring of the toric Deligne-Mumford stack associated to a simplicial toric variety is a flat deformation of (but is not necessarily isomorphic to) the Chow ring of a crepant resolution.

Flops of G-Hilb and equivalences of derived categories by variation of GIT quotient

For a finite subgroup G⊂SL(3,ℂ), Bridgeland, King, and Reid [BKR] proved that the moduli space of G-clusters is a crepant resolution of the quotient ℂ3/G . This paper considers the moduli spaces $\mathcal{M}$θ, introduced by Kronheimer and further studied by Sardo Infirri, which coincide with G-Hilb for a particular choice of geometric invariant theory (GIT) parameter θ. For G Abelian, we prove that every projective crepant resolution of ℂ3/G is isomorphic to $\mathcal{M}$θ for some parameter θ. The key step is the description of GIT chambers in terms of the K-theory of the moduli space via the appropriate Fourier-Mukai transform. We also uncover explicit equivalences between the derived categories of moduli $\mathcal{M}$θ for parameters lying in adjacent GIT chambers.

Coinvariant Algebras of Finite Subgroups of SL(3;C)

AbstractFor most of the finite subgroups of SL(3; C) we give explicit formulae for the Molien series of the coinvariant algebras, generalizing McKay's formulae [McKay99] for subgroups of SU(2). We also study the G-orbit Hilbert scheme HilbG(C3) for any finite subgroup G of SO(3), which is known to be a minimal (crepant) resolution of the orbit space C3/G. In this case the fiber over the origin of the Hilbert-Chow morphism from HilbG(C3) to C3/G consists of finitely many smooth rational curves, whose planar dual graph is identified with a certain subgraph of the representation graph of G. This is an SO(3) version of the McKay correspondence in the SU(2) case.

Representations of symplectic reflection algebras and resolutions of deformations of symplectic quotient singularities

We give an equivalence of triangulated categories between the derived category of finitely generated representations of symplectic reflection algebras associated with wreath products (with parameter t=0) and the derived category of coherent sheaves on a crepant resolution of the spectrum of the centre of these algebras.

Uniqueness of crepant resolutions and symplectic singularities

Nous démontrons l'unicité des résolutions crépantes pour certaines singularités quotient et pour certaines adhérences d'orbites nilpotentes. La finitude des résolutions symplectiques non-isomorphes pour les singularités symplectiques de dimension 4 est démontrée. Nous construisons aussi un exemple d'une singularité symplectique qui admet deux résolutions symplectiques non-équivalentes.

On crepant resolutions of symplectic quotient singularities

On crepant resolutions of symplectic quotient singularities

Orbifold Cohomology as Periodic Cyclic Homology

It is known from the work of Feigin–Tsygan, Weibel and Keller that the cohomology groups of a smooth complex variety X can be recovered from (roughly speaking) its derived category of coherent sheaves. In this paper we show that for a finite group G acting on X the same procedure applied to G-equivariant sheaves gives the orbifold cohomology of X/G.As an application, in some cases we are able to obtain simple proofs of an additive isomorphism between the orbifold cohomology of X/G and the usual cohomology of its crepant resolution (the equality of Euler and Hodge numbers was obtained earlier by various authors). We also state some conjectures on the product structures, as well as the singular case; and a connection with a recent work by Kawamata.

G2quivers

We present, in explicit matrix representation and a modernity befitting the community, the classification of the finite discrete subgroups of G_2 and compute the McKay quivers arising therefrom. Of physical interest are the classes of N=1 gauge theories descending from M-theory and of mathematical interest are possible steps toward a systematic study of crepant resolutions to smooth G_2 manifolds as well as generalised McKay Correspondences. This writing is a companion monograph to hep-th/9811183 and hep-th/9905212, wherein the analogues for Calabi-Yau three- and four-folds were considered.