Crepant Resolution Research Articles

Deformations of some local Calabi-Yau manifolds

We study deformations of certain crepant resolutions of isolated rational Gorenstein singularities. After a general discussion of the deformation theory, we specialize to dimension $3$ and consider examples which are good (log) resolutions as well as the case of small resolutions. We obtain some partial results on the classification of canonical threefold singularities that admit good crepant resolutions. Finally, we study a noncrepant example, the blowup of a small resolution whose exceptional set is a smooth curve.

A Kummer construction for Chern–Ricci flat balanced manifolds

Given a non-Kähler Calabi–Yau compact orbifold with isolated singularities endowed with a Chern–Ricci flat balanced metric, we study, via a gluing construction, the existence of Chern–Ricci flat balanced metrics on its crepant resolutions, and discuss applications to the search of solutions for the Hull–Strominger system. We also describe the scenario of singular threefolds with ordinary double points, and see that similarly is possible to obtain balanced approximately Chern–Ricci flat metrics.

Spherical objects in dimensions two and three

This paper classifies spherical objects in various geometric settings in dimensions two and three, including both minimal and partial crepant resolutions of Kleinian singularities, as well as arbitrary flopping 3 -fold contractions with only Gorenstein terminal singularities. The main result is much more general: in each such setting, we prove that all objects in the associated null category \mathcal{C} with no negative Ext groups are the image, under the action of an appropriate braid or pure braid group, of some object in the heart of a bounded t-structure. The corollary is that all objects x which admit no negative Exts, and for which \mathrm{Hom}_{\mathcal{C}}(x,x)=\mathbb{C} , are the images of the simples. A variation on this argument goes further and classifies all bounded t-structures on \mathcal{C} . There are multiple geometric, topological and algebraic consequences, primarily to autoequivalences and stability conditions. Our main new technique also extends into representation theory, and we establish that in the derived category of a finite-dimensional algebra which is silting discrete, every object with no negative Ext groups lies in the heart of a bounded t-structure. As a consequence, every semibrick complex can be completed to a simple-minded collection.

Crepant Resolutions of Stratified Varieties via Gluing

Abstract Let $X$ be a variety with a stratification ${\mathcal{S}}$ into smooth locally closed subvarieties such that $X$ is locally a product along each stratum (e.g., a symplectic singularity). We prove that assigning to each open subset $U \subset X$ the set of isomorphism classes of locally projective crepant resolutions of $U$ defines an ${\mathcal{S}}$-constructible sheaf of sets. Thus, for each stratum $S$ and basepoint $s \in S$, the fundamental group acts on the set of germs of projective crepant resolutions at $s$, leaving invariant the germs extending to the entire stratum. Global locally projective crepant resolutions correspond to compatible such choices for all strata. For example, if the local projective crepant resolutions are unique, they automatically glue uniquely. We give criteria for a locally projective crepant resolution $\rho : \tilde X \to X$ to be globally projective. We show that the sheafification of the presheaf $U \mapsto \operatorname{Pic}(\rho ^{-1}(U)/U)$ of relative Picard classes is also constructible. The resolution is globally projective only if there exist local relatively ample bundles whose classes glue to a global section of this sheaf. The obstruction to lifting this section to a global ample line bundle is encoded by a gerbe on the singularity $X$. We show the gerbes are automatically trivial if $X$ is a symplectic quotient singularity. Our main results hold in the more general setting of partial crepant resolutions, that need not have smooth source. We apply the theory to symmetric powers and Hilbert schemes of surfaces with du Val singularities, finite quotients of tori, multiplicative, and Nakajima quiver varieties, as well as to canonical three-fold singularities.

Mutations of noncommutative crepant resolutions in geometric invariant theory

Let X be a generic quasi-symmetric representation of a connected reductive group G. The GIT quotient stack X=[Xss(ℓ)/G]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {X}=[X^\ ext {ss}(\\ell )/G]$$\\end{document} with respect to a generic ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell $$\\end{document} is a (stacky) crepant resolution of the affine quotient X/G, and it is derived equivalent to a noncommutative crepant resolution (=NCCR) of X/G. Halpern-Leistner and Sam showed that the derived category Db(cohX)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\ extrm{D}}^{\ extrm{b}}}({\ ext {coh}}\\mathfrak {X})$$\\end{document} is equivalent to certain subcategories of Db(coh[X/G])\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\ extrm{D}}^{\ extrm{b}}}({\ ext {coh}}[X/G])$$\\end{document}, which are called magic windows. This paper studies equivalences between magic windows that correspond to wall-crossings in a hyperplane arrangement in terms of NCCRs. We show that those equivalences coincide with derived equivalences between NCCRs induced by tilting modules, and that those tilting modules are obtained by certain operations of modules, which is called exchanges of modules. When G is a torus, it turns out that the exchanges are nothing but iterated Iyama–Wemyss mutations. Although we mainly discuss resolutions of affine varieties, our theorems also yield a result for projective Calabi-Yau varieties. Using techniques from the theory of noncommutative matrix factorizations, we show that Iyama–Wemyss mutations induce a group action of the fundamental group π1(P1\\{0,1,∞})\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pi _1(\\mathbb {P}^1\\,\\backslash \\{0,1,\\infty \\})$$\\end{document} on the derived category of a Calabi-Yau complete intersection in a weighted projective space.

Non-commutative resolutions of linearly reductive quotient singularities

ABSTRACT We prove the existence of non-commutative crepant resolutions (in the sense of Van den Bergh) of quotient singularities by finite and linearly reductive group schemes in positive characteristic. In dimension 2, we relate these to resolutions of singularities provided by G-Hilbert schemes and F-blowups. As an application, we establish and recover results concerning resolutions for toric singularities, as well as canonical, log terminal and F-regular singularities in dimension 2.

Construction and deformations of Calabi–Yau 3-folds in codimension 4

We construct polarized Calabi–Yau 3-folds with at worst isolated canonical orbifold points in codimension 4 that can be described in terms of the equations of the Segre embedding of P2×P2 in P8. We investigate the existence of other deformation families in their Hilbert scheme by either studying Tom and Jerry degenerations or by comparing their Hilbert series with those of existing low codimension Calabi–Yau 3-folds. Among other interesting results, we find a family of Calabi–Yau 3-fold with five distinct Tom and Jerry deformation families, a phenomenon not seen for Q-Fano 3-folds. We compute the Hodge numbers of P2×P2 Calabi–Yau 3-folds and corresponding manifolds obtained by performing crepant resolutions. We obtain a manifold with a pair of Hodge numbers that does not appear in the famously known list of 30108 distinct Hodge pairs of Kruzer–Skarke, in the list of 7890 distinct Hodge pairs corresponding to complete intersections in the product of projective spaces and in Hodge paris obtained from Calabi–Yau 3-folds having low codimension embeddings in weighted projective spaces.

Singular contact varieties

In this note we propose the generalization of the notion of a holomorphic contact structure on a manifold (smooth variety) to varieties with rational singularities and prove basic properties of such objects. Natural examples of singular contact varieties come from the theory of nilpotent orbits: every projectivization of the closure of a nilpotent orbit in a semisimple Lie algebra satisfies our definition after normalization. We show the correspondence between symplectic varieties with the structure of a C∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {C}^*$$\\end{document}-bundle and the contact ones along with the existence of stratification à la Kaledin. In the projective case we demonstrate the equivalence between crepant and contact resolutions of singularities, show the uniruledness and give a full classification of projective contact varieties in dimension 3.

3d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 theories from M-theory on CY4 and IIB brane box

We study 3D N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 supersymmetric field theories geometrically engineered from M-theory on non-compact Calabi-Yau fourfolds (CY4). We establish a detailed dictionary between the geometry and topology of non-compact CY4 and the physics of 3D N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 theories in three different regimes. The first one is the Coulomb branch description when the CY4 is smooth. The second one is non-abelian gauge theory when the CY4 has a degenerate ℙ1-fibration structure. The third one is the strongly coupled SCFT from a CY4 singularity. We find interesting flavor symmetry enhancements in the singular limit of CY4, as well as an interesting and previously unexplored phenomenon in 3D, termed “flavor symmetry duality”. Many examples are analyzed with an emphasis on toric CY4s and ℂ4 orbifolds with crepant resolutions. We develop a new brane box method to study the physics of Coulomb branch of 3D N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 theory that admits a toric construction. Via IIB/M-theory duality we find that the brane box diagram living in ℝ3 can be physically realized as a configuration of intersecting 4-branes which are extended objects in 8D maximal supersymmetric theory, which is shown to be consistent via various chains of dualities. The rank, effective gauge coupling and certain hints to flavor symmetry enhancement of the 3D N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 theory are read off from the brane box and cross-checked against the results obtained from geometric engineering. The exotic branes in 8D maximal supersymmetric theory and the 4-string junctions thereof are shown to play a crucial role in the construction of the brane box.

All 81 crepant resolutions of a finite quotient singularity are hyperpolygon spaces

We demonstrate that the linear quotient singularity for the exceptional subgroup G G in S p ( 4 , C ) \mathrm {Sp}(4,\mathbb {C}) of order 32 is isomorphic to an affine quiver variety for a 5-pointed star-shaped quiver. This allows us to construct uniformly all 81 projective crepant resolutions of C 4 / G \mathbb {C}^4/G as hyperpolygon spaces by variation of GIT quotient, and we describe both the movable cone and the Namikawa Weyl group action via an explicit hyperplane arrangement. More generally, for the n n -pointed star shaped quiver, we describe completely the birational geometry for the corresponding hyperpolygon spaces in dimension 2 n − 6 2n-6 ; for example, we show that there are 1684 projective crepant resolutions when n = 6 n=6 . We also prove that the resulting affine cones are not quotient singularities for n ≥ 6 n \geq 6 .

Topological Strings on Non-commutative Resolutions

In this paper we propose a definition of torsion refined Gopakumar–Vafa (GV) invariants for Calabi–Yau threefolds with terminal nodal singularities that do not admit Kähler crepant resolutions. Physically, the refinement takes into account the charge of five-dimensional BPS states under a discrete gauge symmetry in M-theory. We propose a mathematical definition of the invariants in terms of the geometry of all non-Kähler crepant resolutions taken together. The invariants are encoded in the A-model topological string partition functions associated to non-commutative (nc) resolutions of the Calabi–Yau. Our main example will be a singular degeneration of the generic Calabi–Yau double cover of P3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {P}}^3$$\\end{document} and leads to an enumerative interpretation of the topological string partition function of a hybrid Landau–Ginzburg model. Our results generalize a recent physical proposal made in the context of torus fibered Calabi–Yau manifolds by one of the authors and clarify the associated enumerative geometry.

5d SCFTs from isolated complete intersection singularities

In this paper, we explore the zoo of 5d superconformal field theories (SCFTs) constructed from M-theory on Isolated Complete Intersection Singularities (ICIS). We systematically investigate the crepant resolution of such singularities, and obtain a classification of rank ⩽ 10 models with a smooth crepant resolution and smooth exceptional divisors, as well as a number of infinite sequences with the same smoothness properties. For these models, we study their Coulomb branch properties and compute the flavor symmetry algebra from the resolved CY3 and/or the magnetic quiver. We check the validity of the conjectures relating the properties of the 5d SCFT and the 4d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 SCFT from IIB superstring on the same singularity. When the 4d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 SCFT has a Lagrangian quiver gauge theory description, one can obtain the magnetic quiver of the 5d theory by gauging flavor symmetry, which encodes the 5d Higgs branch information. Regarding the smoothness of the crepant resolution and integrality of 4d Coulomb branch spectrum, we find examples with a smooth resolved CY3 and smooth exceptional divisors, but fractional 4d Coulomb branch spectrum. Moreover, we compute the discrete (higher)-symmetries of the 5d/4d SCFTs from the link topology for a few examples.

Crepant Resolutions, Mutations, and the Space of Potentials

The McKay correspondence has had much success in studying resolutions of 3-fold quotient singularities through a wide range of tools coming from geometry, combinatorics, and representation theory. We develop a computational perspective in this setting primarily realized through a web application to explore mutations of quivers with potential and crepant triangulations. We use this to study flops between different crepant resolutions of Gorenstein toric quotient singularities and find many situations in which the mutations of a quiver with potential classifies them. The application also implements key constructions of the McKay correspondence, including the Craw–Reid procedure and the process of associating a quiver to a toric resolution.

Sixfolds of generalized Kummer type and K3 surfaces

We prove that any hyper-Kähler sixfold $K$ of generalized Kummer type has a naturally associated manifold $Y_K$ of $\mathrm {K}3^{[3]}$ type. It is obtained as crepant resolution of the quotient of $K$ by a group of symplectic involutions acting trivially on its second cohomology. When $K$ is projective, the variety $Y_K$ is birational to a moduli space of stable sheaves on a uniquely determined projective $\mathrm {K}3$ surface $S_K$. As an application of this construction we show that the Kuga–Satake correspondence is algebraic for the K3 surfaces $S_K$, producing infinitely many new families of $\mathrm {K}3$ surfaces of general Picard rank $16$ satisfying the Kuga–Satake Hodge conjecture.

Steady Kähler–Ricci solitons on crepant resolutions of finite quotients of Cn$\mathbb {C}^n$

AbstractWe prove the existence of steady Kähler–Ricci solitons on equivariant crepant resolutions of , where is a finite subgroup of acting freely on .

A Donaldson-Thomas crepant resolution conjecture on Calabi-Yau 4-folds

Let G G be a finite subgroup of S U ( 4 ) \mathrm {SU}(4) such that its elements have age at most one. In the first part of this paper, we define K K -theoretic stable pair invariants on a crepant resolution of the affine quotient C 4 / G {\mathbb {C}}^4/G , and conjecture a closed formula for their generating series in terms of the root system of G G . In the second part, we define degree zero Donaldson-Thomas invariants of Calabi-Yau 4-orbifolds, develop a vertex formalism that computes the invariants in the toric case, and conjecture closed formulae for their generating series for the quotient stacks [ C 4 / Z r ] [{\mathbb {C}}^4/{\mathbb {Z}}_r] , [ C 4 / Z 2 × Z 2 ] [{\mathbb {C}}^4/{\mathbb {Z}}_2\times {\mathbb {Z}}_2] . Combining these two parts, we formulate a crepant resolution correspondence which relates the above two theories.

Contact invariants of [formula omitted]-Gorenstein toric contact manifolds, the Ehrhart polynomial and Chen-Ruan cohomology

Q-Gorenstein toric contact manifolds provide an interesting class of examples of contact manifolds with torsion first Chern class. They are completely determined by certain rational convex polytopes, called toric diagrams, and arise both as links of toric isolated singularities and as prequantizations of monotone toric symplectic orbifolds. In this paper we show how the cylindrical contact homology invariants of a Q-Gorenstein toric contact manifold are related to•the Ehrhart (quasi-)polynomial of its toric diagram;•the Chen-Ruan cohomology of any crepant toric orbifold resolution of its corresponding toric isolated singularity;•the Chen-Ruan cohomology of any monotone toric symplectic orbifold base that gives rise to it through prequantization.

Gromov-Witten invariants of 𝑆𝑦𝑚^{𝑑}ℙ^{𝕣}

We give a graph-sum algorithm that expresses any genus- g g Gromov-Witten invariant of the symmetric product orbifold Sym d ⁡ P r ≔ \operatorname {Sym}^d\mathbb {P}^r≔ [ ( P r ) d / S d ] [(\mathbb {P}^r)^d/S_d] in terms of “Hurwitz-Hodge integrals” – integrals over (compactified) Hurwitz spaces. We apply the algorithm to prove a mirror-type theorem for Sym d ⁡ P r \operatorname {Sym}^d\mathbb {P}^r in genus zero. The theorem states that a generating function of Gromov-Witten invariants of Sym d ⁡ P r \operatorname {Sym}^d\mathbb {P}^r is equal to an explicit power series I Sym d ⁡ P r I_{\operatorname {Sym}^d\mathbb {P}^r} , conditional upon a conjectural combinatorial identity. This is a first step in the direction of proving Ruan’s Crepant Resolution Conjecture for the resolution H i l b ( d ) ( P 2 ) Hilb^{(d)}(\mathbb {P}^2) of the coarse moduli space of Sym d ⁡ P 2 . \operatorname {Sym}^d\mathbb {P}^2.

A note on affine cones over Grassmannians and their stringy 𝐸-functions

We compute the stringy E E -function of the affine cone over a Grassmannian. If the Grassmannian is not a projective space then its cone does not admit a crepant resolution. Nonetheless the stringy E E -function is sometimes a polynomial and in those cases the cone admits a noncommutative crepant resolution. This raises the question as to whether the existence of a noncommutative crepant resolution implies that the stringy E E -function is a polynomial.

On the M2–Brane Index on Noncommutative Crepant Resolutions

On certain M-theory backgrounds which are a circle fibration over a smooth Calabi–Yau the quantum theory of M2 branes can be studied in terms of the K-theoretic Donaldson–Thomas theory on the threefold. We extend this relation to noncommutative crepant resolutions. In this case the threefold develops a singularity and classical smooth geometry is replaced by the algebra of paths of a certain quiver. K-theoretic quantities on the quiver representation moduli space can be computed via toric localization and result in certain rational functions of the toric parameters. We discuss in particular the case of the conifold and certain orbifold singularities.