Stability conditions and crepant small resolutions
Yukinobu Toda
备注：
A lot of changes. Section 5 and Section 6 in the previous version are removed, and some explanations for the background are added. The statement in the original version contains a mistake, and added an assumption for a crepant small resolution. To apper in Transactions of the American Mathematical Society
In this paper, we describe the spaces of stability conditions on the triangulated categories associated to three dimensional crepant small resolutions. The resulting spaces have chamber structures such that each chamber corresponds to a birational model together with a special Fourier-Mukai transform. We observe these spaces are covering spaces over certain open subsets of finite dimensional vector spaces, and determine their deck transformations.
In this paper, we show a condition for two-parameter Gorenstein cyclic quotient singularities to have a crepant resolution by using the remainder polynomial in any dimension.
We compute open GW invariants for $\mathcal{K}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1}$, open orbifold GW invariants for $[\C^3/\Z_2]$, formulate an open crepant resolution conjecture and verify it for this pair. We show that open invariants can be glued together to deduce the Bryan-Graber closed crepant resolution conjecture for the orbifold $[\mathcal{O}_{\mathbb{P}^1}(-1)\oplus\mathcal{O}_{\mathbb{P}^1}(-1)/\Z_2]$ and its crepant resolution $\mathcal{K}_{\mathbb{P}^1\times\mathbb{P}^1}$.
We prove the uniqueness of crepant resolutions for some quotient singularities and for some nilpotent orbits. The finiteness of non-isomorphic symplectic resolutions for 4-dimenensional symplectic singularities is proved. We also give an example of symplectic singularity which admits two non-equivalent symplectic resolutions.
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.
We show a sufficient condition for Fujiki-Oka resolutions of Gorenstein abelian quotient singularities to be crepant in all dimensions by using Ashikaga's continuous fractions. Moreover, we prove that all three dimensional Gorenstein abelian quotient singularities possess a crepant resolution as a corollary. This alternative proof of existence needs only simple computations comparing with the results ever known.
In characteristic zero, if a quotient variety has a crepant resolution, the Euler characteristic of the crepant resolution is equal to the number of conjugacy classes of the acting group, by Batyrev's theorem. This is one of the McKay correspondence. It is natural to consider the analogue statement in the positive characteristic. In this paper, we present sequences of crepant resolutions of quotient varieties in the positive characteristic and show that one of the sequences gives a counterexample to the analogue statement of Batyrev's theorem.
One of our results of this article is that every (projective) crepant resolution of a Slodowy slice in a nilpotent orbit closure in $\mathfrak{sl}_N(\mathbb{C})$ can be obtained as the restriction of some crepant resolution of the nilpotent orbit closure. We also show that there is a decomposition of the Slodowy slice into other Slodowy slices with good properties. From this decomposition, one can count the number of crepant resolutions.
We study the relationship between Gromov-Witten invariants of local $\mathbb{P}^4$ and Gromov-witten invariants of $[\mathbb{C}^5/\mathbb{Z}_5]$ for all genera. We state the crepant resolution conjecture in explicit form and prove this conjecture for $g=2,3.$
We study the relation among the genus 0 Gromov-Witten theories of the three spaces $\mathcal{X}\leftarrow\mathcal{Z}\leftarrow Y$, where $\mathcal{X}=[\c^2/\z_3]$, $\mathcal{Z}$ is obtained by a weighted blowup at the stacky point of $\mathcal{X}$, and $Y$ is the crepant resolution of the $A_2$ singularity. We formulate and verify a statement similar to the Crepant Resolution Conjecture of Bryan and Graber.b
We study Cox rings of crepant resolutions of quotient singularities $\mathbb{C}^3/G$ where $G$ is a finite subgroup of $SL(3,\mathbb{C})$. We use them to obtain information on the geometric structure of these resolutions, number of different resolutions and relations between them. In particular, we treat explicitly several examples where $G$ contains elements of age 2.
We prove the cohomological crepant resolution conjecture of Ruan for the weighted projective space P(1,3,4,4). To compute the quantum corrected cohomology ring we combine the results of Coates-Corti-Iritani-Tseng on P(1,1,1,3) and our previous results.
We study resolutions of singularities of orbit closures in quiver representations. We consider certain resolutions of singularities which have already been constructed by Reineke, and we determine under which conditions they are crepant. More specifically, we deal with quivers of type A3, An, D4. Then, as an application of the results we have found, we construct several Fano and Calabi-Yau varieties as orbital degeneracy loci.
Let $V$ be a complex vector space on which a finite group $G$ acts by linear transformations. Let $W = V \oplus V^*$ be the sum of $V$ with its dual $V^*$. We prove that if the quotient $W/G$ admits a smooth crepant resolution, then the subgroup $G \subset Aut V$ is generated by complex reflections. We also obtain some results on the structure of smooth crepant resolutions of the quotients $W/G$, where $W$ is a symplectic vector space, and $G \subset Aut W$ is a finite group of symplectic linear transformations of the vector space $W$.
We compare the Chen-Ruan cohomology ring of the weighted projective spaces $\IP(1,3,4,4)$ and $\IP(1,...,1,n)$ with the cohomology ring of their crepant resolutions. In both cases, we prove that the Chen-Ruan cohomology ring is isomorphic to the quantum corrected cohomology ring of the crepant resolution after suitable evaluation of the quantum parameters. For this, we prove a formula for the Gromov-Witten invariants of the resolution of a transversal ${\rm A}_3$ singularity.
After fixing a non-degenerate bilinear form on a vector space V we define an involution of the manifold of flags F in V by taking a flag to its orthogonal complement. When V is of dimension 3 we check that the Crepant Resolution Conjecture of J. Bryan and T. Graber holds: the genus zero (orbifold) Gromov-Witten potential function of [F / Z_2] agrees (up to unstable terms) with the genus zero Gromov-Witten potential function of a crepant resolution Y of the quotient scheme F / Z_2, after setting a quantum parameter to -1, making a linear change of variables, and analytically continuing coeffici…
For an arbitrary integer $r\geq 1$, we compute $r$-framed motivic PT and DT invariants of small crepant resolutions of toric Calabi-Yau $3$-folds, establishing a "higher rank" version of the motivic DT/PT wall-crossing formula. This generalises the work of Morrison and Nagao. Our formulae, in particular their relationship with the $r=1$ theory, fit nicely in the current development of higher rank refined DT invariants.
We study variations of tautological bundles on moduli spaces of representations of quivers with relations associated with dimer models under a change of stability parameters. We prove that if the tautological bundle induces a derived equivalence for some stability parameter, then the same holds for any generic stability parameter, and any projective crepant resolution can be obtained as the moduli space for some stability parameter. This result is used in 0905.0059 to prove the abelian McKay correspondence without using the result of math/9908027.
In this article, we determine the explicit toric variety structure of $\hl^{A_1(n)}(\CZ^n)$ for $n=4,5$, where $A_1(n)$ is the special diagonal group of all order 2 elements. Through the toric data of $\hl^{A_1(n)}(\CZ^n)$, we obtain certain toric crepant resolutions of $\CZ^n/A_1(n)$, and the different crepant resolutions are connected by flops of $n$-folds for $n=4,5$.
We state a version of the crepant resolution conjecture for total ancestor potentials for surface singularities, and reduce the conjecture to the quantum McKay correspondence conjecture of J.Bryan and A.Gholampour and a vanishing conjecture for Hurwitz-Hodge integrals. In particular, for singularities of type A, we prove the conjecture. We also suggest an approach towards a proof for the general cases by Teleman's reconstruction theorem for semisimple cohomological field theories.
We study the orbifold Gromov-Witten theory of the quotient C^3/Z_3 in all genera. Our first result is a proof of the holomorphic anomaly equations in the precise form predicted by B-model physics. Our second result is an exact crepant resolution correspondence relating the Gromov-Witten theories of C^3/Z_3 and local CP2. The proof of the correspondence requires an identity proven in the Appendix by T. Coates and H. Iritani.
For orbifolds admitting a crepant resolution and satisfying a hard Lefschetz condition, we formulate a conjectural equivalence between the Gromov-Witten theories of the orbifold and the resolution. We prove the conjecture for the equivariant Gromov-Witten theories of the nth symmetric product of the complex plane and the Hilbert scheme of n points in the plane.
For a finite abelian subgroup $G\subset SL_n(\mathbb{C})$, we study whether a given crepant resolution $X$ of the quotient variety $\mathbb{C}^n/G$ is obtained as a moduli space of $G$-constellations. In particular we show that, if $X$ admits a natural $G$-constellation family in the sense of Logvinenko over it with all fibers being indecomposable as $\mathbb{C}[\mathbb{C}^n]$-modules, then $X$ is isomorphic to the normalization of a fine moduli space of $G$-constellations.
Let G be the group A_4 or Z_2xZ_2. We compute the integral of λ_g on the Hurwitz locus H_G\subset M_g of curves admitting a degree 4 cover of P^1 having monodromy group G. We compute the generating functions for these integrals and write them as a trigonometric expression summed over the positive roots of the E_6 and D_4 root systems respectively. As an application, we prove the Crepant Resolution Conjecture for the orbifolds [C^3/A_4] and [C^3/(Z_2xZ_2)].
We prove a comparison formula for curve-counting invariants in the setting of the McKay correspondence, related to the crepant resolution conjecture for Donaldson-Thomas invariants. The conjecture is concerned with comparing the invariants of a (hard Lefschetz) Calabi-Yau orbifold of dimension three with those of a specific crepant resolution of its coarse moduli space. We prove the conjecture for point classes and give a conditional proof for general curve classes. We also prove a variant of the formula for curve classes. Along the way we identify the image of the standard heart of the orbifo…
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.
We prove the crepant resolution conjecture for Donaldson-Thomas invariants of hard Lefschetz CY3 orbifolds, formulated by Bryan-Cadman-Young, interpreting the statement as an equality of rational functions. In order to do so, we show that the generating series of stable pair invariants on any CY3 orbifold is the expansion of a rational function. As a corollary, we deduce a symmetry of this function induced by the derived dualising functor. Our methods also yield a proof of the orbifold DT/PT correspondence for multi-regular curve classes on hard Lefschetz CY3 orbifolds.
Let J be an abelian surface with a generic ample line bundle O(1). For n>0, the moduli space M(2, 0, 2n) of O(1)-semistable sheaves F of rank 2 with Chern classes c_1(F) = 0, c_2(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 M(2; 0; 2n) for n>1. This certainly implies that there is no symplectic desingularization of M(2, 0, 2n) for n>1.
An immediate generalization of the classical McKay correspondence for Gorenstein quotient spaces $\Bbb{C}^{r}/G$ in dimensions $r\geq 4$ would primarily demand the existence of projective, crepant, full desingularizations. Since this is not always possible, it is natural to ask about special classes of such quotient spaces which would satisfy the above property. In this paper we give explicit necessary and sufficient conditions under which 2-parameter series of Gorenstein cyclic quotient singularities have torus-equivariant resolutions of this specific sort in all dimensions.
Given any smooth toric surface S, we prove a SYM-HILB correspondence which relates the 3-point, degree zero, extended Gromov-Witten invariants of the n-fold symmetric product stack [Sym^n(S)] of S to the 3-point extremal Gromov-Witten invariants of the Hilbert scheme Hilb^n(S) of n points on S. As we do not specialize the values of the quantum parameters involved, this result proves a strengthening of Ruan's Cohomological Crepant Resolution Conjecture for the Hilbert-Chow morphism from Hilb^n(S) to Sym^n(S) and yields a method of reconstructing the cup product for Hilb^n(S) from the orbifold i…
We prove that the tangent developables of the varieties appearing in the third row of the Tits-Freudenthal magic square admit categorical crepant resolutions of singularities.
A collection $S = \{D_1,\ldots, D_n\}$ of divisors in a smooth variety $X$ is an {\em arrangement} if intersections of all subsets of $S$ are smooth. We show that a double cover of $X$ ramified on an arrangement has a crepant resolution under additional hypotheses. Namely, we assume that all intersection components that change the canonical divisor when blown up satisfy are {\em splayed}, a property of the tangent spaces of the components first studied by Faber. This strengthens a result of Cynk and Hulek, which requires a stronger hypothesis on the intersection components. Further, we study t…
Given a brane tiling, that is, a bipartite graph on a torus, we can associate with it a singular 3-Calabi-Yau variety. In this paper we study its commutative and non-commutative crepant resolutions. We give an explicit toric description of all its commutative crepant resolutions. We also explain how the McKay correspondence in dimension 3 can be interpreted using brane tilings.
Using the theory of dimer models Broomhead proved that every 3-dimensional Gorenstein affine toric variety Spec R admits a toric non-commutative crepant resolution (NCCR). We give an alternative proof of this result by constructing a tilting bundle on a (stacky) crepant resolution of Spec R using standard toric methods. Our proof does not use dimer models.
Given a brane tiling, that is, a bipartite graph on a torus, we can associate with it a singular 3-Calabi-Yau variety. Using the brane tiling, we can also construct all crepant resolutions of the above variety. We give an explicit toric description of tilting bundles on these crepant resolutions. This result proves the conjecture of Hanany, Herzog and Vegh and a version of the conjecture of Aspinwall.
Motivated by physics, we propose two conjectures regarding the cohomology ring of the crepant resolutions of orbifolds and cohomological invariants of K-equivalent manifolds.
In this paper we analyze four examples of birational transformations between local Calabi-Yau 3-folds: two crepant resolutions, a crepant partial resolution, and a flop. We study the effect of these transformations on genus-zero Gromov-Witten invariants, proving the Coates-Corti-Iritani-Tseng/Ruan form of the Crepant Resolution Conjecture in each case. Our results suggest that this form of the Crepant Resolution Conjecture may also hold for more general crepant birational transformations. They also suggest that Ruan's original Crepant Resolution Conjecture should be modified, by including appr…
A formula for calculating the Lefschetz number of an automorphism acting on a crepant resolution for a quotient of a Kahler manifold derived from an equivariant version of McKay correspondence. The latter is proven in some cases. As an application the Lefschetz numbers of of involutions acting on Calabi-Yau threefolds and their mirrors are compared.
Let $X$ be an algebraic variety with Gorenstein singularities. We define the notion of a wonderful resolution of singularities of $X$ by analogy with the theory of wonderful compactifications of semi-simple linear algebraic groups. We prove that if $X$ has rational singularities and has a wonderful resolution of singularities, then $X$ admits a categorical crepant resolution of singularities. As an immediate corollary, we get that all determinantal varieties defined by the minors of a generic square/symmetric/skew-symmetric matrix admit categorical crepant resolution of singularities. We also …
The Abuaf-Ueda flop is a 7-dimensional flop related to $G_2$ homogeneous spaces. The derived equivalence for this flop was first proved by Ueda using mutations of semi-orthogonal decompositions. In this article, we give an alternative proof for the derived equivalence using tilting bundles. Our proof also shows the existence of a non-commutative crepant resolution of the singularity appearing in the flopping contraction. We also give some results on moduli spaces of finite-length modules over this non-commutative crepant resolution.
We give a criterion for the existence of non-commutative crepant resolutions (NCCR's) for certain toric singularities. In particular we recover Broomhead's result that a 3-dimensional toric Gorenstein singularity has a NCCR. Our result also yields the existence of a NCCR for a 4-dimensional toric Gorenstein singularity which is known to have no toric NCCR.
We prove an all genera version of the Crepant Resolution Conjecture of Ruan and Bryan-Graber for type A surface singularities. We are based on a method that explicitly computes Hurwitz-Hodge integrals described in an earlier paper and some recent results by Liu-Xu for some intersection numbers on the Deligne-Mumford moduli spaces. We also generalize our results to some three-dimensional orbifolds.
We prove that a crepant resolution of a Ricci-flat Kähler cone X admits a complete Ricci-flat Kähler metric asymptotic to the cone metric in every Kähler class in H^2_c(Y,R). This result contains as a subcase the existence of ALE Ricci-flat Kähler metrics on crepant resolutions of X=C^n /G, where G is a finite subgroup of SL(n,C). We consider the case in which X is toric. A result of A. Futaki, H. Ono, and G. Wang guarantees the existence of a Ricci-flat Kähler cone metric if X is Gorenstein. We use toric geometry to construct crepant resolutions.
