Crepant Resolution Research Articles

Dimer models and Calabi-Yau algebras

In this article we study dimer models, as introduced in string theory, which give a way of writing down a class of non-commutative `superpotential' algebras. Some examples are 3-dimensional Calabi-Yau algebras, as defined by Ginzburg, and some are not. We consider two types of `consistency' condition on dimer models, and show that a `geometrically consistent' model is `algebraically consistent'. We prove that the algebra obtained from an algebraically consistent dimer model is a 3-dimensional Calabi-Yau algebra and finally prove that this gives a non-commutative crepant resolution of the Gorenstein affine toric threefold associated to the dimer model.

Refined open noncommutative Donaldson–Thomas invariants for small crepant resolutions

The aim of this paper is to study an analog of non-commutative Donaldson-Thomas theory corresponding to the refined topological vertex for small crepant resolutions of toric Calabi-Yau 3-folds. We define the invariants using dimer models and provide "wall-crossing" formulas. In particular, we get normalized generating functions which are unchanged under "wall-crossing".

Cellular resolutions of noncommutative toric algebras from superpotentials

This paper constructs cellular resolutions for classes of noncommutative algebras, analogous to those introduced by Bayer and Sturmfels (1998) [2] in the commutative case. To achieve this we generalise the dimer model construction of noncommutative crepant resolutions of three-dimensional toric algebras by associating a superpotential and a notion of consistency to toric algebras of arbitrary dimension. For abelian skew group algebras and algebraically consistent dimer model algebras, we introduce a cell complex Δ in a real torus whose cells describe uniformly all maps in the minimal projective bimodule resolution of A. We illustrate the general construction of Δ for an example in dimension four arising from a tilting bundle on a smooth toric Fano threefold to highlight the importance of the incidence function on Δ.

Picard groups of punctured spectra of dimension three local hypersurfaces are torsion-free

AbstractLet (R,m) be a Noetherian local ring and UR=Spec(R)−{m} be the punctured spectrum of R. Gabber conjectured that if R is a complete intersection of dimension three, then the abelian group Pic(UR) is torsion-free. In this note we prove Gabber’s statement for the hypersurface case. We also point out certain connections between Gabber’s conjecture, Van den Bergh’s notion of non-commutative crepant resolutions and some well-studied questions in homological algebra over local rings.

THE GEOMETRY AND ARITHMETIC OF A CALABI–YAU SIEGEL THREEFOLD

In this paper we treat in details a Siegel modular variety [Formula: see text] that has a Calabi–Yau model, [Formula: see text]. We shall describe the structure of the ring of modular forms and its geometry. We shall illustrate two different methods of producing the Hodge numbers. The first uses the definition of [Formula: see text] as the quotient of another known Calabi–Yau variety [Formula: see text]. In this case we will get the Hodge numbers considering the action of the group on a crepant resolution [Formula: see text] of [Formula: see text]. The second, purely algebraic geometric, uses the equations derived from the ring of modular forms and is based on determining explicitly the Calabi–Yau model [Formula: see text] and computing the Picard group and the Euler characteristic.

The orbifold topological vertex

We develop a topological vertex formalism for computing the Donaldson–Thomas invariants of Calabi–Yau orbifolds. The basic combinatorial object is the orbifold vertex V λ μ ν G , a generating function for the number of 3D partitions asymptotic to 2D partitions λ, μ, ν and colored by representations of a finite Abelian group G acting on C 3 . In the case where G ≅ Z n acting on C 3 with transverse A n − 1 quotient singularities, we give an explicit formula for V λ μ ν G in terms of Schur functions. We discuss applications of our formalism to the Donaldson–Thomas crepant resolution conjecture and to the orbifold Donaldson–Thomas/Gromov–Witten correspondence. We also explicitly compute the Donaldson–Thomas partition function for some simple orbifold geometries: the local football P a , b 1 and the local B Z 2 gerbe.

Quantum moduli space of Chern-Simons quivers, wrapped D6-branes and AdS 4 /CFT 3

We study the quantum moduli space of $ \mathcal{N} = 2 $ Chern-Simons quivers with generic ranks and CS levels, proving along the way exact formulas for the charges of bare monopole operators. We then derive $ \mathcal{N} = 2 $ Chern-Simons quiver theories dual to $ Ad{S_4} \times {Y^{p,q}}\left( {\mathbb{C}{\mathbb{P}^2}} \right) $ M-theory backgrounds, for the whole family of Sasaki-Einstein seven-manifolds and for any value of the torsion G 4 flux. The derivation of the gauge theories relies on the reduction to type IIA string theory, in which M2-branes become D2-branes while the conical geometry maps to RR flux and D6-branes wrapped on compact four-cycles. M5-branes on torsion cycles map to flux and wrapped D4-branes. The moduli space of the quiver is shown to contain the corresponding CY4 cone and all its crepant resolutions.

Generalized Borcea–Voisin Construction

C. Voisin and C. Borcea have constructed mirror pairs of families of Calabi-Yau threefolds by taking the quotient of the product of an elliptic curve with a K3 surface endowed with a non-symplectic involution. In this paper, we generalize the construction of Borcea and Voisin to any prime order and build three and four dimensional Calabi-Yau orbifolds. We classify the topological types that are obtained and show that, in dimension 4, orbifolds built with an involution admit a crepant resolution and come in topological mirror pairs. We show that for odd primes, there are generically no minimal resolutions and the mirror pairing is lost.

Hyperconifold transitions, mirror symmetry, and string theory

Multiply-connected Calabi–Yau threefolds are of particular interest for both string theorists and mathematicians. Recently it was pointed out that one of the generic degenerations of these spaces (occurring at codimension one in moduli space) is an isolated singularity which is a finite cyclic quotient of the conifold; these were called hyperconifolds. It was also shown that if the order of the quotient group is even, such singular varieties have projective crepant resolutions, which are therefore smooth Calabi–Yau manifolds. The resulting topological transitions were called hyperconifold transitions, and change the fundamental group as well as the Hodge numbers. Here Batyrevʼs construction of Calabi–Yau hypersurfaces in toric fourfolds is used to demonstrate that certain compact examples containing the remaining hyperconifolds — the Z 3 and Z 5 cases — also have Calabi–Yau resolutions. The mirrors of the resulting transitions are studied and it is found, surprisingly, that they are ordinary conifold transitions. These are the first examples of conifold transitions with mirrors which are more exotic extremal transitions. The new hyperconifold transitions are also used to construct a small number of new Calabi–Yau manifolds, with small Hodge numbers and fundamental group Z 3 or Z 5 . Finally, it is demonstrated that a hyperconifold is a physically sensible background in Type IIB string theory. In analogy to the conifold case, non-perturbative dynamics smooth the physical moduli space, such that hyperconifold transitions correspond to non-singular processes in the full theory.

Topological properties of Hilbert schemes of almost-complex four-manifolds II

In this article, we study the rational cohomology rings of Voisin's punctual Hilbert schemes $X^{[n]}$ associated to a symplectic compact fourfold $X$. We prove that these rings can be universally constructed from $H^*(X,\mathbb{Q})$ and $c_1(X)$, and that Ruan's crepant resolution conjecture holds if $c_1(X)$ is a torsion class. Next, we prove that for any almost-complex compact fourfold $X$, the complex cobordism class of $X^{[n]}$ depends only on the cobordism class of $X$.

Poisson deformations of affine symplectic varieties

We shall prove that the Poisson deformation functor of an affine (singular) symplectic variety is unobstructed. As a corollary, we prove the following result. Theorem: For an affine symplectic variety $X$ with a good $C^*$-action (where its natural Poisson structure is positively weighted), the following are equivalent (1) $X$ has a crepant projective resolution. (2) $X$ has a smoothing by a Poisson deformation.

The genus zero Gromov–Witten invariants of the symmetric square of the plane

We study the Abramovich–Vistoli moduli space of genus zero orbifold stable maps to [Sym P], the stack symmetric square of P. This space compactifies the moduli space of stable maps from hyperelliptic curves to P, and we show that all genus zero Gromov–Witten invariants are determined from trivial enumerative geometry of hyperelliptic curves. We also show how the genus zero Gromov–Witten invariants can be used to determine the number of hyperelliptic curves of degree d and genus g interpolating 3d+ 1 generic points in P. Comparing our method to that of Graber for calculating the same numbers, we verify an example of the crepant resolution conjecture.

Counting Invariant of Perverse Coherent Sheaves and its Wall-crossing

We introduce moduli spaces of stable perverse coherent systems on small crepant resolutions of Calabi–Yau 3-folds and consider their Donaldson–Thomas-type counting invariants. The stability depends on the choice of a component ( = a chamber) in the complement of finitely many lines ( = walls) in the plane. We determine all walls and compute generating functions of invariants for all choices of chambers when the Calabi–Yau is the resolved conifold. For suitable choices of chambers, our invariants are specialized to Donaldson–Thomas, Pandharipande–Thomas and Szendroi invariants.

Hyperbolic geometry and non-Kähler manifolds with trivial canonical bundle

We use hyperbolic geometry to construct simply-connected symplectic or complex manifolds with trivial canonical bundle and with no compatible Kahler structure. We start with the desingularisations of the quadric cone in C^4: the smoothing is a natural S^3-bundle over H^3, its holomorphic geometry is determined by the hyperbolic metric; the small-resolution is a natural S^2-bundle over H^4 with symplectic geometry determined by the metric. Using hyperbolic geometry, we find orbifold quotients with trivial canonical bundle; smooth examples are produced via crepant resolutions. In particular, we find the first example of a simply-connected symplectic 6-manifold with c_1=0 that does not admit a compatible Kahler structure. We also find infinitely many distinct complex structures on 2(S^3xS^3)#(S^2xS^4) with trivial canonical bundle. Finally, we explain how an analogous construction for hyperbolic manifolds in higher dimensions gives symplectic non-Kahler "Fano" manifolds of dimension 12 and higher.

Non-commutative desingularization of determinantal varieties I

We show that determinantal varieties defined by maximal minors of a generic matrix have a non-commutative desingularization, in that we construct a maximal Cohen-Macaulay module over such a variety whose endomorphism ring is Cohen-Macaulay and has finite global dimension. In the case of the determinant of a square matrix, this gives a non-commutative crepant resolution.

Generating functions for colored 3D Young diagrams and the Donaldson-Thomas invariants of orbifolds

We derive two multivariate generating functions for three-dimensional Young diagrams (also called plane partitions). The variables correspond to a colouring of the boxes according to a finite Abelian subgroup G of SO(3). We use the vertex operator methods of Okounkov--Reshetikhin--Vafa for the easy case G = Z/n; to handle the considerably more difficult case G=Z/2 x Z/2, we will also use a refinement of the author's recent q--enumeration of pyramid partitions. In the appendix, we relate the diagram generating functions to the Donaldson-Thomas partition functions of the orbifold C^3/G. We find a relationship between the Donaldson-Thomas partition functions of the orbifold and its G-Hilbert scheme resolution. We formulate a crepant resolution conjecture for the Donaldson-Thomas theory of local orbifolds satisfying the Hard Lefschetz condition.

Remarks on non-commutative crepant resolutions of complete intersections

We study obstructions to existence of non-commutative crepant resolutions, in the sense of Van den Bergh, over local complete intersections.

Poisson deformations of affine symplectic varieties, II

This is a continuation of math.AG/0609741. Let Y be an affine symplectic variety with a C^*-action with positive weights, and let \pi: X -> Y be its crepant resolution. Then \pi induces a natural map PDef(X) -> PDef(Y) of Kuranishi spaces for the Poisson deformations of X and Y. In the Part I, we proved that PDef(X) and PDef(Y) are both non-singular, and this map is a finite surjective map. In this paper (Part II), we prove that it is a Galois covering. Markman already obtained a similar result in the compact case, which was a motivation of this paper. As an application, we shall construct explicitly the universal Poisson deformation of the normalization \tilde{O} of a nilpotent orbit closure \bar{O} in a complex simple Lie algebra when \tilde{O} has a crepant resolution.

Examples of asymptotically conical Ricci-flat Kähler manifolds

The author has proved that a crepant resolution Y of a Ricci-flat K\"{a}hler cone X admits a complete Ricci-flat K\"{a}hler metric asymptotic to the cone metric in every K\"{a}hler class in H^2_c(Y,\R). These manifolds are generalizations of the Ricci-flat ALE K\"{a}hler spaces known by the work of P. Kronheimer, D. Joyce and others. This article considers further the problem of constructing examples. We show that every 3-dimensional Gorenstein toric K\"{a}hler cone admits a crepant resolution for which the above theorem applies. This gives infinitely many examples of asymptotically conical Ricci-flat manifolds. Then other examples are given of which are crepant resolutions hypersurface singularities which are known to admit Ricci-flat K\"{a}hler cone metrics by the work of C. Boyer, K. Galicki, J. Koll\'{a}r, and others. Two families of hypersurface examples are given which are distinguished by the condition b_3(Y)=0 or b_3(Y)>0.

Ricci-flat Kähler metrics on crepant resolutions of Kähler cones

We prove that a crepant resolution of a Ricci-flat K\"ahler cone X admits a complete Ricci-flat K\"ahler metric asymptotic to the cone metric in every K\"ahler class in H^2_c(Y,R). This result contains as a subcase the existence of ALE Ricci-flat K\"ahler 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 K\"ahler cone metric if X is Gorenstein. We use toric geometry to construct crepant resolutions.