Crepant Resolution Research Articles

Applications of homological mirror symmetry to hypergeometric systems: Duality conjectures

Homological mirror symmetry for crepant resolutions of Gorenstein toric singularities leads to a pair of conjectures on certain hypergeometric systems of PDEs. We explain these conjectures and verify them in some cases.

Noncommutative (Crepant) Desingularizations and the Global Spectrum of Commutative Rings

In this paper we study endomorphism rings of finite global dimension over not necessarily normal commutative rings. These objects have recently attracted attention as noncommutative (crepant) resolutions, or NC(C)Rs, of singularities. We propose a notion of a NCCR over any commutative ring that appears weaker but generalizes all previous notions. Our results yield strong necessary and sufficient conditions for the existence of such objects in many cases of interest. We also give new examples of NCRs of curve singularities, regular local rings and normal crossing singularities. Moreover, we introduce and study the global spectrum of a ring R, that is, the set of all possible finite global dimensions of endomorphism rings of MCM R-modules. Finally, we use a variety of methods to compute global dimension for many endomorphism rings.

Geometric Reid’s recipe for dimer models

Crepant resolutions of three-dimensional toric Gorenstein singularities are derived equivalent to noncommutative algebras arising from consistent dimer models. By choosing a special stability parameter and hence a distinguished crepant resolution $$Y$$ , this derived equivalence generalises the Fourier-Mukai transform relating the $$G$$ -Hilbert scheme and the skew group algebra $$\mathbb {C}[x,y,z]*G$$ for a finite abelian subgroup of $$\mathrm{SL }(3,\mathbb {C})$$ . We show that this equivalence sends the vertex simples to pure sheaves, except for the zero vertex which is mapped to the dualising complex of the compact exceptional locus. This generalises results of Cautis–Logvinenko (J Reine Angew Math 636:193–236, 2009) and Cautis–Craw–Logvinenko (J Reine Angew Math arXiv:1205.3110 , 2014) to the dimer setting, though our approach is different in each case. We also describe some of these pure sheaves explicitly and compute the support of the remainder, providing a dimer model analogue of results from Logvinenko (J Algebra 324:2064–2087, 2010).

Automorphism group of Batyrev Calabi–Yau threefolds

In this paper, we will prove that all Batyrev Calabi–Yau threefolds, arising from a crepant resolution of a generic hyperplane section of a toric Fano–Gorenstein fourfold, have finite automorphism group. Together with the Morrison conjecture, this suggests that all Batyrev Calabi–Yau threefolds should have a polyhedral Kahler (ample) cone.

The -cyclic McKay correspondence via motivic integration

AbstractWe study the McKay correspondence for representations of the cyclic group of order $p$ in characteristic $p$. The main tool is the motivic integration generalized to quotient stacks associated to representations. Our version of the change of variables formula leads to an explicit computation of the stringy invariant of the quotient variety. A consequence is that a crepant resolution of the quotient variety (if any) has topological Euler characteristic $p$ as in the tame case. Also, we link a crepant resolution with a count of Artin–Schreier extensions of the power series field with respect to weights determined by ramification jumps and the representation.

Existence of complete Kähler Ricci-flat metrics on crepant resolutions

In this note, we obtain existence results for complete Ricci-flat Kähler metrics on crepant resolutions of singularities of Calabi–Yau varieties. Furthermore, for certain asymptotically flat Calabi–Yau varieties, we show that the Ricci-flat metric on the resolved manifold has the same asymptotic behavior as the initial variety.

Box graphs and singular fibers

We determine the higher codimension fibers of elliptically fibered Calabi-Yau fourfolds with section by studying the three-dimensional N=2 supersymmetric gauge theory with matter which describes the low energy effective theory of M-theory compactified on the associated Weierstrass model, a singular model of the fourfold. Each phase of the Coulomb branch of this theory corresponds to a particular resolution of the Weierstrass model, and we show that these have a concise description in terms of decorated box graphs based on the representation graph of the matter multiplets, or alternatively by a class of convex paths on said graph. Transitions between phases have a simple interpretation as `flopping' of the path, and in the geometry correspond to actual flop transitions. This description of the phases enables us to enumerate and determine the entire network between them, with various matter representations for all reductive Lie groups. Furthermore, we observe that each network of phases carries the structure of a (quasi-)minuscule representation of a specific Lie algebra. Interpreted from a geometric point of view, this analysis determines the generators of the cone of effective curves as well as the network of flop transitions between crepant resolutions of singular elliptic Calabi-Yau fourfolds. From the box graphs we determine all fiber types in codimensions two and three, and we find new, non-Kodaira, fiber types for E_6, E_7 and E_8.

Determinantal Quintics and Mirror Symmetry of Reye Congruences

We study a certain family of determinantal quintic hypersurfaces in $\mathbb{P}^{4}$ whose singularities are similar to the well-studied Barth-Nieto quintic. Smooth Calabi-Yau threefolds with Hodge numbers $(h^{1,1},h^{2,1})=(52,2)$ are obtained by taking crepant resolutions of the singularities. It turns out that these smooth Calabi-Yau threefolds are in a two dimensional mirror family to the complete intersection Calabi-Yau threefolds in $\mathbb{P}^{4}\times\mathbb{P}^{4}$ which have appeared in our previous study of Reye congruences in dimension three. We compactify the two dimensional family over $\mathbb{P}^{2}$ and reproduce the mirror family to the Reye congruences. We also determine the monodromy of the family over $\mathbb{P}^{2}$ completely. Our calculation shows an example of the orbifold mirror construction with a trivial orbifold group.

Lagrangian Floer Superpotentials and Crepant Resolutions for Toric Orbifolds

We investigate the relationship between the Lagrangian Floer superpotentials for a toric orbifold and its toric crepant resolutions. More specifically, we study an open string version of the crepant resolution conjecture (CRC) which states that the Lagrangian Floer superpotential of a Gorenstein toric orbifold $\mathcal{X}$ and that of its toric crepant resolution $Y$ coincide after analytic continuation of quantum parameters and a change of variables. Relating this conjecture with the closed CRC, we find that the change of variable formula which appears in closed CRC can be explained by relations between open (orbifold) Gromov-Witten invariants. We also discover a geometric explanation (in terms of virtual counting of stable orbi-discs) for the specialization of quantum parameters to roots of unity which appears in Y. Ruan's original CRC ["The cohomology ring of crepant resolutions of orbifolds", Gromov-Witten theory of spin curves and orbifolds, 117-126, Contemp. Math., 403, Amer. Math. Soc., Providence, RI, 2006]. We prove the open CRC for the weighted projective spaces $\mathcal{X}=\mathbb{P}(1,\ldots,1,n)$ using an equality between open and closed orbifold Gromov-Witten invariants. Along the way, we also prove an open mirror theorem for these toric orbifolds.

Counting hyperelliptic curves on an Abelian surface with quasi-modular forms

In this paper we produce a generating function for the number of hyperelliptic curves (up to translation) on a polarized Abelian surface using the crepant resolution conjecture and the Yau-Zaslow formula. We present a formula to compute these in terms of MacMahon’s generalized sum-of-divisors functions, and prove that they are quasi-modular forms.

A McKay-like correspondence for $(0, 2)$-deformations

We present a local computation of deformations of the tangent bundle for a resolved orbifold singularity C d =G. These correspond to (0; 2)-deformations of (2; 2)-theories. A McKay-like correspondence is found predicting the dimension of the space of rst-order deformations from simple calculations involving the group. This is conrmed in two dimensions using the Kronheimer{Nakajima quiver construction. In higher dimensions such a computation is subject to nontrivial worldsheet instanton corrections and some examples are given where this happens. However, we conjecture that the special crepant resolution given by the GHilbert scheme is never subject to such corrections, and show this is true in an innite number of cases. Amusingly, for three-dimensional examples where G is abelian, the moduli space is associated to a quiver given by the toric fan of the blow-up. It is shown that an orbifold of the form C 3 =Z7 has a nontrivial superpotential and thus an obstructed moduli space.

On semi-continuity problems for minimal log discrepancies

Abstract We show the semi-continuity property of minimal log discrepancies for varieties which have a crepant resolution in the category of Deligne–Mumford stacks. Using this property, we also prove the ideal-adic semi-continuity problem for toric pairs.

Wonderful resolutions and categorical crepant resolutions of singularities

Abstract 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.

Maximal modifications and Auslander–Reiten duality for non-isolated singularities

We first generalize classical Auslander–Reiten duality for isolated singularities to cover singularities with a one-dimensional singular locus. We then define the notion of CT modules for non-isolated singularities and we show that these are intimately related to noncommutative crepant resolutions (NCCRs). When R has isolated singularities, CT modules recover the classical notion of cluster tilting modules but in general the two concepts differ. Then, wanting to generalize the notion of NCCRs to cover partial resolutions of $\operatorname{Spec}R$ , in the main body of this paper we introduce a theory of modifying and maximal modifying modules. Under mild assumptions all the corresponding endomorphism algebras of the maximal modifying modules for three-dimensional Gorenstein rings are shown to be derived equivalent. We then develop a theory of mutation for modifying modules which is similar but different to mutations arising in cluster tilting theory. Our mutation works in arbitrary dimension, and in dimension three the behavior of our mutation strongly depends on whether a certain factor algebra is artinian.

A geometric definition of Gabrielov numbers

Gabrielov numbers describe certain Coxeter–Dynkin diagrams of the 14 exceptional unimodal singularities and play a role in Arnold’s strange duality. In a previous paper, the authors defined Gabrielov numbers of a cusp singularity with an action of a finite abelian subgroup $$G$$ of $$\mathrm{SL}(3,{\mathbb {C}})$$ using the Gabrielov numbers of the cusp singularity and data of the group $$G$$ . Here we consider a crepant resolution $$Y \rightarrow {\mathbb {C}}^3/G$$ and the preimage $$Z$$ of the image of the Milnor fibre of the cusp singularity under the natural projection $${\mathbb {C}}^3 \rightarrow {\mathbb {C}}^3/G$$ . Using the McKay correspondence, we compute the homology of the pair $$(Y,Z)$$ . We construct a basis of the relative homology group $$H_3(Y,Z;{\mathbb {Q}})$$ with a Coxeter–Dynkin diagram where one can read off the Gabrielov numbers.

Asymptotically conical Calabi–Yau manifolds, I

This is the first part in a two-part series on complete Calabi-Yau manifolds asymptotic to Riemannian cones at infinity. We begin by proving general existence and uniqueness results. The uniqueness part relaxes the decay condition $O(r^{-n-\epsilon})$ needed in earlier work to $O(r^{-\epsilon})$, relying on some new ideas about harmonic functions. We then look at a few examples: (1) Crepant resolutions of cones. This includes a new class of Ricci-flat small resolutions associated with flag manifolds. (2) Affine deformations of cones. One focus here is the question of the precise rate of decay of the metric to its tangent cone. We prove that the optimal rate for the Stenzel metric on $T^*S^n$ is $-2\frac{n}{n-1}$.

On singular fibres in F-theory

In this paper, we propose a connection between the field theory local model (Katz-Vafa field theory) and the type of singular fibre in flat crepant resolutions of elliptic Calabi-Yau fourfolds, a class of fourfolds considered by Esole and Yau. We review the analysis of degenerate fibres for models with gauge groups SU(5) and SO(10) in detail, and observe that the naively expected fibre type is realized if and only if the Higgs vev in the field theory local model is unramified. To test this idea, we implement a linear (unramified) Higgs vev for the `E6' Yukawa point in a model with gauge group SU(5) and verify that this indeed leads to a fibre of Kodaira type IV*. Based on this observation, we argue i) that the singular fibre types appearing in the fourfolds studied by Esole-Yau are not puzzling at all, (so that this class of fourfolds does not have to be excluded from the candidate of input data of some yet-unknown formulation of F-theory) and ii) that such fourfold geometries also contain more information than just the eigenvalues of the Higgs field vev configuration in the field theory local models.

DEGENERATIONS OF RICCI-FLAT CALABI–YAU MANIFOLDS

This paper is a sequel to [Continuity of extremal transitions and flops for Calabi–Yau manifolds, J. Differential Geom.89 (2011) 233–270]. We further investigate the Gromov–Hausdorff convergence of Ricci-flat Kähler metrics under degenerations of Calabi–Yau manifolds. We extend Theorem 1.1 in [Continuity of extremal transitions and flops for Calabi–Yau manifolds, J. Differential Geom.89 (2011) 233–270] by removing the condition on existence of crepant resolutions for Calabi–Yau varieties.

Tate form and weak coupling limits in F-theory

We consider the weak coupling limit of F-theory in the presence of non-Abelian gauge groups implemented using the traditional ansatz coming from Tate's algorithm. We classify the types of singularities that could appear in the weak coupling limit and explain their resolution. In particular, the weak coupling limit of SU(n) gauge groups leads to an orientifold theory which suffers from conifold singulaties that do not admit a crepant resolution compatible with the orientifold involution. We present a simple resolution to this problem by introducing a new weak coupling regime that admits singularities compatible with both a crepant resolution and an orientifold symmetry. We also comment on possible applications of the new limit to model building. We finally discuss other unexpected phenomena as for example the existence of several non-equivalent directions to flow from strong to weak coupling leading to different gauge groups.

The diversity of symplectic Calabi-Yau 6-manifolds

Given an integer b and a finitely presented group G, we produce a compact symplectic 6-manifold with c1=0, b2>b, b3>b and π1=G. In the simply connected case, we can also arrange for b3=0; in particular, these examples are not diffeomorphic to Kähler manifolds with c1=0. The construction begins with a certain orientable, four-dimensional, hyperbolic orbifold assembled from right-angled 120-cells. The twistor space of the hyperbolic orbifold is a symplectic Calabi–Yau orbifold; a crepant resolution of this last orbifold produces a smooth symplectic manifold with the required properties.