Crepant Resolution Research Articles

Non-commutative crepant resolutions for some toric singularities. II

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.

Holography, matrix factorizations and K-stability

Placing D3-branes at conical Calabi-Yau threefold singularities produces many AdS5/CFT4 duals. Recent progress in differential geometry has produced a technique (called K-stability) to recognize which singularities admit conical Calabi-Yau metrics. On the other hand, the algebraic technique of non-commutative crepant resolutions, involving matrix factorizations, has been developed to associate a quiver to a singularity. In this paper, we put together these ideas to produce new AdS5/CFT4 duals, with special emphasis on non-toric singularities.

Birational geometry of symplectic quotient singularities

For a finite subgroup Gamma subset mathrm {SL}(2,mathbb {C}) and for nge 1, we use variation of GIT quotient for Nakajima quiver varieties to study the birational geometry of the Hilbert scheme of n points on the minimal resolution S of the Kleinian singularity mathbb {C}^2/Gamma . It is well known that X:={{,mathrm{{mathrm {Hilb}}},}}^{[n]}(S) is a projective, crepant resolution of the symplectic singularity mathbb {C}^{2n}/Gamma _n, where Gamma _n=Gamma wr mathfrak {S}_n is the wreath product. We prove that every projective, crepant resolution of mathbb {C}^{2n}/Gamma _n can be realised as the fine moduli space of theta -stable Pi -modules for a fixed dimension vector, where Pi is the framed preprojective algebra of Gamma and theta is a choice of generic stability condition. Our approach uses the linearisation map from GIT to relate wall crossing in the space of theta -stability conditions to birational transformations of X over mathbb {C}^{2n}/Gamma _n. As a corollary, we describe completely the ample and movable cones of X over mathbb {C}^{2n}/Gamma _n, and show that the Mori chamber decomposition of the movable cone is determined by an extended Catalan hyperplane arrangement of the ADE root system associated to Gamma by the McKay correspondence. In the appendix, we show that morphisms of quiver varieties induced by variation of GIT quotient are semismall, generalising a result of Nakajima in the case where the quiver variety is smooth.

Non-Commutative Crepant Resolutions for Some Toric Singularities I

Abstract We give a criterion for the existence of noncommutative crepant resolutions (NCCRs) for certain toric singularities. In particular, we recover Broomhead’s result that a three-dimensional toric Gorenstein singularity has an NCCR. Our result also yields the existence of an NCCR for a four-dimensional toric Gorenstein singularity, which is known to have no toric NCCR.

Incidence geometry in a Weyl chamber II: SLn

We study the polyhedral geometry of the hyperplanes orthogonal to the weights of the first and the second fundamental representations of sln inside the dual fundamental Weyl chamber. We obtain generating functions that enumerate the flats and the faces of a fixed dimension. In addition, we describe the extreme rays of the incidence geometry and classify simplicial faces.From the perspective of supersymmetric gauge theories with 8 supercharges in five-dimensional spacetime, the poset of flats is isomorphic to the network of mixed Coulomb-Higgs branches. On the other hand, the poset of faces is conjectured to be isomorphic to the network of crepant partial resolutions of an elliptic fibration with gauge algebra sln and “matter representation” given by the sum of the first two fundamental representations.

Modular curves and Mordell-Weil torsion in F-theory

In this work we prove a bound for the torsion in Mordell-Weil groups of smooth elliptically fibered Calabi-Yau 3- and 4-folds. In particular, we show that the set of torsion groups which can occur on a smooth elliptic Calabi-Yau n-fold is contained in the set of subgroups which appear on a rational elliptic surface if n ≥ 3 and is slightly larger for n = 2. The key idea in our proof is showing that any elliptic fibration with sufficiently large torsion1 has singularities in codimension 2 which do not admit a crepant resolution. We prove this by explicitly constructing and studying maps to a modular curve whose existence is predicted by a universal property. We use the geometry of modular curves to explain the minimal singularities that appear on an elliptic fibration with prescribed torsion, and to determine the degree of the fundamental line bundle (hence the Kodaira dimension) of the universal elliptic surface which we show to be consistent with explicit Weierstrass models. The constraints from the modular curves are used to bound the fundamental group of any gauge group G in a supergravity theory obtained from F-theory. We comment on the isolated 8-dimensional theories, obtained from extremal K3’s, that are able to circumvent lower dimensional bounds. These theories neither have a heterotic dual, nor can they be compactified to lower dimensional minimal SUGRA theories. We also comment on the maximal, discrete gauged symmetries obtained from certain Calabi-Yau threefold quotients.

Gorenstein modifications and \mathds{𝑄}-Gorenstein rings

Let R R be a Cohen–Macaulay normal domain with a canonical module ω R \omega _R . It is proved that if R R admits a noncommutative crepant resolution (NCCR), then necessarily it is Q \mathds {Q} -Gorenstein. Writing S S for a Zariski local canonical cover of R R , a tight relationship between the existence of noncommutative (crepant) resolutions on R R and S S is given. A weaker notion of Gorenstein modification is developed, and a similar tight relationship is given. There are three applications: non-Gorenstein quotient singularities by connected reductive groups cannot admit an NCCR, the centre of any NCCR is log-terminal, and the Auslander–Esnault classification of two-dimensional CM-finite algebras can be deduced from Buchweitz–Greuel–Schreyer.

Conic divisorial ideals of Hibi rings and their applications to non-commutative crepant resolutions

In this paper, we study divisorial ideals of a Hibi ring which is a toric ring arising from a partially ordered set. We especially characterize the special class of divisorial ideals called conic using the associated partially ordered set. Using our description of conic divisorial ideals, we also construct a module giving a non-commutative crepant resolution (NCCR) of the Segre product of polynomial rings. Furthermore, applying the operation called mutation, we give other modules giving NCCRs of it.

On Ruan’s cohomological crepant resolution conjecture for the complexified Bianchi orbifolds

We give formulae for the Chen-Ruan orbifold cohomology for the orbifolds given by a Bianchi group acting on complex hyperbolic 3-space. The Bianchi groups are the arithmetic groups PSL\_2(A), where A is the ring of integers in an imaginary quadratic number field. The underlying real orbifolds which help us in our study, given by the action of a Bianchi group on real hyperbolic 3-space (which is a model for its classifying space for proper actions), have applications in physics.We then prove that, for any such orbifold, its Chen-Ruan orbifold cohomology ring is isomorphic to the usual cohomology ring of any crepant resolution of its coarse moduli space.By vanishing of the quantum corrections, we show that this result fits in with Ruan's Cohomological Crepant Resolution Conjecture.

Euler Characteristics of Crepant Resolutions of Weierstrass Models

Based on an identity of Jacobi, we prove a simple formula that computes the pushforward of analytic functions of the exceptional divisor of a blowup of a projective variety along a smooth complete intersection with normal crossing. We apply this pushforward formula to derive generating functions for Euler characteristics of crepant resolutions of singular Weierstrass models given by Tate's algorithm. Since these Euler characteristics depend only on the sequence of blowups and not on the Kodaira fiber itself, nor the associated group, several distinct Tate models have the same Euler characteristic. In the case of elliptic Calabi-Yau threefolds, we also compute the Hodge numbers. For elliptically fibered Calabi-Yau fourfolds, our results also prove a conjecture of Blumenhagen-Grimm-Jurke-Weigand based on F-theory/heterotic string duality.

The Hilb/Sym correspondence for $${\mathbb {C}}^2$$: descendents and Fourier–Mukai

We study here the crepant resolution correspondence for the $$\mathsf {T}$$ -equivariant descendent Gromov–Witten theories of $$\mathsf {Hilb}^n({\mathbb {C}}^2)$$ and $$\mathsf {Sym}^n({\mathbb {C}}^2)$$ . The descendent correspondence is obtained from our previous matching of the associated CohFTs by applying Givental’s quantization formula to a specific symplectic transformation $$\mathsf {K}$$ . The first result of the paper is an explicit computation of $$\mathsf {K}$$ . Our main result then establishes a fundamental relationship between the Fourier–Mukai equivalence of the associated derived categories (by Bridgeland, King, and Reid) and the symplectic transformation $$\mathsf {K}$$ via Iritani’s integral structure. The results use Haiman’s Fourier–Mukai calculations and are exactly aligned with Iritani’s point of view on crepant resolution.

Hori-mological projective duality

Kuznetsov has conjectured that Pfaffian varieties should admit noncommutative crepant resolutions which satisfy his Homological Projective Duality. We prove half the cases of this conjecture by interpreting and proving a duality of nonabelian gauged linear sigma models proposed by Hori.

Noncommutative quasi-resolutions

The notion of a noncommutative quasi-resolution is introduced for a noncommutative noetherian algebra with singularities, even for a non-Cohen-Macaulay algebra. If A is a commutative normal Gorenstein domain, then a noncommutative quasi-resolution of A naturally produces a noncommutative crepant resolution (NCCR) of A in the sense of Van den Bergh, and vice versa. Under some mild hypotheses, we prove that(i)in dimension two, all noncommutative quasi-resolutions of a given noncommutative algebra are Morita equivalent, and(ii)in dimension three, all noncommutative quasi-resolutions of a given noncommutative algebra are derived equivalent. These assertions generalize important results of Van den Bergh, Iyama-Reiten and Iyama-Wemyss in the commutative and central-finite cases.

Generalized Orbifold Euler Characteristics on the Grothendieck Ring of Varieties with Actions of Finite Groups

The notion of the orbifold Euler characteristic came from physics at the end of the 1980s. Coincidence (up to sign) of the orbifold Euler characteristics is a necessary condition for crepant resolutions of orbifolds to be mirror symmetric. There were defined higher order versions of the orbifold Euler characteristic and generalized (“motivic”) versions of them. In a previous paper, the authors defined a notion of the Grothendieck ring K 0 fGr ( Var C ) of varieties with actions of finite groups on which the orbifold Euler characteristic and its higher order versions are homomorphisms to the ring of integers. Here, we define the generalized orbifold Euler characteristic and higher order versions of it as ring homomorphisms from K 0 fGr ( Var C ) to the Grothendieck ring K 0 ( Var C ) of complex quasi-projective varieties and give some analogues of the classical Macdonald equations for the generating series of the Euler characteristics of the symmetric products of a space.

Crepant resolutions of [formula omitted] and the generalized Kronheimer construction (in view of the gauge/gravity correspondence)

As a continuation of a general program started in two previous publications, in the present paper we study the Kähler quotient resolution of the orbifold ℂ3∕Z4, comparing with the results of a toric description of the same. In this way we determine the algebraic structure of the exceptional divisor, whose compact component is the second Hirzebruch surface F2. We determine the explicit Kähler geometry of the smooth resolved manifold Y, which is the total space of the canonical bundle of F2. We study in detail the chamber structure of the space of stability parameters (corresponding in gauge theory to the Fayet–Iliopoulos parameters) that are involved in the construction of the desingularizations either by generalized Kronheimer quotient, or as algebro-geometric quotients. The walls of the chambers correspond to two degenerations; one is a partial desingularization of the quotient, which is the total space of the canonical bundle of the weighted projective space P[1,1,2], while the other is the product of the ALE space A1 by a line, and is related to the full resolution in a subtler way. These geometrical results will be used to look for exact supergravity brane solutions and dual superconformal gauge theories.

Non-commutative resolutions of toric varieties

Let R be the coordinate ring of an affine toric variety. We prove, using direct elementary methods, that the endomorphism ring EndR(A), where A is the (finite) direct sum of all (isomorphism classes of) conic R-modules, has finite global dimension equal to the dimension of R. This gives a precise version, and an elementary proof, of a theorem of Špenko and Van den Bergh implying that EndR(A) has finite global dimension. Furthermore, we show that EndR(A) is a non-commutative crepant resolution if and only if the toric variety is simplicial. For toric varieties over a perfect field k of prime characteristic, we show that the ring of differential operators Dk(R) has finite global dimension.

Crepant resolution and the holomorphic anomaly equation for [C3/Z3

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 P 2 . The proof of the correspondence requires an identity proven in the Appendix by T. Coates and H. Iritani.

Homogeneous Contact Manifolds and Resolutions of Calabi–Yau Cones

In the present work we provide a constructive method to describe contact structures on compact homogeneous contact manifolds. The main feature of our approach is to describe the Cartan-Ehresmann connection (gauge field) for principal circle bundles over complex flag manifolds by using elements of representation theory of simple Lie algebras. This description allows us to compute explicitly the expression of the contact form for any Boothby-Wang fibration over complex flag manifolds as well as their induced homogeneous Sasaki-Einstein structures. As an application of our results we use the Cartan-Remmert reduction and the Calabi ansatz technique to provide many explicit examples of crepant resolutions of Calabi-Yau cones with certain homogeneous Sasaki-Einstein manifolds realized as links of isolated hypersurface singularities.

The Geometry of the SU(2) \xd7 G2-model

We study elliptic fibrations that geometrically engineer an SU(2) ×G2 gauge theory realized by a Weierstrass model for the collision III + I0* ns. We find all the distinct crepant resolutions of such a model and the flops connecting them. We compute the generating function for the Euler characteristic of the SU(2) ×G2-model. In the case of a Calabi-Yau threefold, we consider the compactification of M-theory and F-theory on an SU(2) × G2-model to a five and six-dimensional supergravity theory with eight super-charges. By matching each crepant resolution with each Coulomb chamber of the five-dimensional theory, we determine the number of multiplets and compute the prepotential in each Coulomb chamber. In particular, we discuss the counting numbers of hypermultiplets in the presence of singularities. We discuss in detail the cancellation of anomalies of the six-dimensional theory.

Calabi–Yau Quotients of Hyperkähler Four-folds

AbstractThe aim of this paper is to construct Calabi–Yau 4-folds as crepant resolutions of the quotients of a hyperkähler 4-fold $X$ by a non-symplectic involution $\unicode[STIX]{x1D6FC}$. We first compute the Hodge numbers of a Calabi–Yau constructed in this way in a general setting, and then we apply the results to several specific examples of non-symplectic involutions, producing Calabi–Yau 4-folds with different Hodge diamonds. Then we restrict ourselves to the case where $X$ is the Hilbert scheme of two points on a K3 surface $S$, and the involution $\unicode[STIX]{x1D6FC}$ is induced by a non-symplectic involution on the K3 surface. In this case we compare the Calabi–Yau 4-fold $Y_{S}$, which is the crepant resolution of $X/\unicode[STIX]{x1D6FC}$, with the Calabi–Yau 4-fold $Z_{S}$, constructed from $S$ through the Borcea–Voisin construction. We give several explicit geometrical examples of both these Calabi–Yau 4-folds, describing maps related to interesting linear systems as well as a rational $2:1$ map from $Z_{S}$ to $Y_{S}$.