Crepant resolutions and brane tilings II: Tilting bundles
aa r X i v : . [ m a t h . AG ] S e p CREPANT RESOLUTIONS AND BRANE TILINGS II: TILTINGBUNDLES
MARTIN BENDER AND SERGEY MOZGOVOY
Abstract.
Given a brane tiling, that is, a bipartite graph on a torus, we canassociate with it a singular 3-Calabi-Yau variety. Using the brane tiling, wecan also construct all crepant resolutions of the above variety. We give anexplicit toric description of tilting bundles on these crepant resolutions. Thisresult proves the conjecture of Hanany, Herzog and Vegh and a version of theconjecture of Aspinwall. Introduction
The goal of this paper is to prove the conjecture of Hanany, Herzog and Vegh[7] on the description of tilting bundles on the crepant resolutions of singular 3-Calabi-Yau varieties arising from brane tilings. All these crepant resolutions canbe constructed as moduli spaces of representations of some quiver with relations [8,Theorem 15.1]. These moduli spaces are toric 3-Calabi-Yau varieties. An explicitconstruction of their toric diagrams was given in [11].Given a brane tiling, we can associate with it a quiver potential (
Q, W ) anda quiver potential algebra C Q/ ( ∂W ). The singular Calabi-Yau variety mentionedabove is isomorphic to the spectrum of the center of C Q/ ( ∂W ). It has a non-commutative crepant resolution C Q/ ( ∂W ) [3, 11]. Its crepant resolutions aregiven by the moduli spaces M θ = M θ ( C Q/ ( ∂W ) , α ) of θ -semistable C Q/ ( ∂W )-representations of dimension α = (1 , . . . , ∈ Z Q , where θ ∈ Z Q is α -generic. All θ -semistable points in M θ are θ -stable for such θ . Therefore there exists a universal(also called tautological) vector bundle U over M θ , endowed with a structure ofa left C Q/ ( ∂W )-module. It follows from the results of Van den Bergh (see [16])that U is a tilting bundle (an alternative proof can be found in [8]). This vectorbundle can be decomposed into a sum of Q line bundles. We will describe thetoric Cartier divisors inducing these line bundles. Namely, we fix some i ∈ Q andfor every vertex i ∈ Q we choose some path u i : i → i . Intersecting the path u i with perfect matchings (they parametrize 2-dimensional orbits of M θ , i.e. rays ofthe corresponding fan, see Section 2), we get a toric Cartier divisor which inducesa line bundle L i over M θ . We will show that U is isomorphic to the direct sumof L i , i ∈ Q . This description of the tilting bundle was conjectured by Hanany,Herzog and Vegh [7, Section 5.2]. Our result proves also a conjecture of Aspinwall[2] on the existence of some “globally defined” collection of line bundles that givesrise to the tilting collection on M θ for arbitrary generic θ . We should note that asimilar description of the exceptional collections in the context of toric quiver vari-eties (these are moduli spaces of quiver representation for quiver without relations)was given by Altmann and Hille [1]. The paper is organized as follows: In Section 2 we gather preliminary materialon brane tilings and the induced quiver potential algebras. In Section 3 we recallsome results of Thaddeus [15] about toric quotients of toric varieties and provesome facts on the descent of line bundles with respect to such quotients. In Section4 we give a toric description of the tilting bundle on M θ . In Section 5 we give someexplicit examples.We would like to thank Markus Reineke for many useful discussions.2. Preliminaries
Most of the content of this section can be found in [11]. We briefly recall somematerial for the convenience of the reader.A brane tiling is a bipartite graph G = ( G ± , G ) together with an embedding ofthe corresponding CW-complex into the real two-dimensional torus T so that thecomplement T \ G consists of simply-connected components. The set of connectedcomponents of T \ G is denoted by G and is called the set of faces of G .With any brane tiling we can associate a quiver Q = ( Q , Q ) embedded in atorus T and a potential W (linear combination of cycles in Q ), see [12]. The set Q of connected components of T \ Q is called the set of faces of Q . The summandsof W are the cycles along the faces of Q taken with appropriate signs. With thisdata, we associate a quiver potential algebra A = C Q/ ( ∂W ), see [12].For any arrow a ∈ Q , we define s ( a ) , t ( a ) ∈ Q to be its source and target (alsocalled tail and head). Consider a complex of abelian groups Z Q d −→ Z Q d −→ Z Q , where d ( F ) = P a ∈ F a , F ∈ Q and d ( a ) = t ( a ) − s ( a ) for any arrow a ∈ Q . Itshomology groups are isomorphic to the homology groups of the 2-dimensional toruscontaining Q . We define an abelian group Λ by a cocartesian left upper square ofthe following diagram Z Q d ✲ Z Q d ✲ Z Q Z ❄ ω Λ ✲ Λwt ❄ .................. d ✲ Mω M ❄ ......... i ✲ where the left arrow of the square is given by F , F ∈ Q . There exists aunique map d : Λ → Z Q making the right triangle commutative. Let M = ker( d ).There exists a unique map ω M : Z → M making the lower triangle commutative.If G has at least one perfect matching then Λ is a free abelian group and the map ω Λ : Z → Λ is injective (see [12, Lemma 3.3]).We define a weak path in Q to be a path consisting of arrows of Q and theirinverses (for any arrow a , we identify aa − and a − a with trivial paths). For anyweak path u , we define its content | u | ∈ Z Q by counting the arrows of u withappropriate signs. We define the weight of u to be wt( u ) = wt( | u | ) ∈ Λ. We define ω ∈ Λ to be the weight of any cycle along some face of Q . Note that ω = ω Λ (1)and that ω ∈ M . REPANT RESOLUTIONS AND BRANE TILINGS 3
Let B = ker( Z Q → Z ), where the map is given by i i ∈ Q . This group isgenerated by the elements of the form i − j , where i, j ∈ Q . As Q is connected,we conclude that B = im d . There is a short exact sequence0 → M i −→ Λ d −→ B → . One can easily see that rk B = Q −
1, rk Λ = Q + 2, and rk M = 3.Let Λ + ⊂ Λ be a semigroup generated by the weights of the arrows. Let P ⊂ Λ Q be a cone generated by Λ + P = { X a i λ i | a i ∈ Q ≥ , λ i ∈ Λ + for all i } . We define M + = Λ + ∩ M and P M = P ∩ M Q . We do not have Λ + = P ∩ Λ ingeneral. But we have M + = P M ∩ M [11]. This implies that Spec C [ M + ] is anormal toric variety. If the brane tiling is consistent (see e.g. [11]) then the quiverpotential algebra C Q/ ( ∂W ) is a 3-Calabi-Yau algebra (see [12, 4, 5]) and is a non-commutative crepant resolution of Spec C [ M + ] (see [3, 11]). In this paper we willalways assume that our brane tiling is consistent.Let A be the set of all perfect matchings of the bipartite graph G . Any perfectmatching I ∈ A can be considered as a subset of Q . We define a characteristicfunction χ I : Z Q → Z by the rule (for a ∈ Q ) χ I ( a ) = ( , a ∈ I, , a I. For any face F ∈ Q we have χ I ( d ( F )) = 1. Therefore we can consider χ I as alinear map Λ → Z , i.e. as an element χ I ∈ Λ ∨ . We define χ I = i ∗ χ I ∈ M ∨ . Thefamily of all χ I : Λ → Z (resp. χ i : M → Z ) defines a linear map χ : Λ → Z A (resp. χ : M → Z A ).The dual cone P ∨ ∈ Λ ∨ Q is generated by χ I , I ∈ Λ, and all the correspondingrays are extremal in P ∨ [4, Lemma 2.3.4]. Analogously, the dual cone P ∨ M ∈ M ∨ Q is generated by χ I , I ∈ A . We denote by A e ⊂ A the set of perfect matchings I ,such that χ I generates an extremal ray in P ∨ M . These perfect matchings are calledextremal.All the crepant resolutions of Spec C [ M + ] can be described as moduli spacesof stable representations of A = C Q/ ( ∂W ) in the sense of King [10]. Let α =(1 , . . . , ∈ Z Q and let θ ∈ B (i.e. θ ∈ Z Q is such that θ · α = 0). Any A -module X can be described by the set of vector spaces ( X i ) i ∈ Q and linear maps X a : X i → X j for arrows a : i → j . We define dim X = (dim X i ) i ∈ Q ∈ Z Q . An A -module X of dimension α is called θ -semistable (resp. θ -stable) if for any proper A -submodule 0 = Y ⊂ X we have θ · dim Y ≥ θ · dim Y > θ is α -generic if for any 0 < β < α we have θ · β = 0. In this case all θ -semistablemodules of dimension α are automatically stable. One can construct the modulispace M θ = M θ ( A, α ) of θ -semistable A -modules of dimension α [10]. It is shownin [11] that, for α -generic θ ∈ B , this moduli space is a toric variety (with a densesubtorus T M = Hom Z ( M, C ∗ ), see Section 3) M θ = Spec C [Λ + ] // θ T B = Spec C [ P ∩ Λ] // θ T B , where T B = Hom Z ( B, C ∗ ). In this case M θ is smooth and is a crepant resolutionof Spec C [ M + ] (see [9, 11]). MARTIN BENDER AND SERGEY MOZGOVOY
An explicit description of a fan of M θ was given in [11]. For any A -module X = (( X i ) i ∈ Q , ( X a ) a ∈ Q ), we define its cosupport I X = { a ∈ Q | X a = 0 } . It wasshown in [11] that every T M -orbit of M θ is uniquely determined by the cosupport ofits modules. Any such cosupport I can be considered as a subgraph of the bipartitegraph G . It was shown in [11] that this subgraph can have at most one connectedcomponent containing more than one edge (we call it a big component of I ). Proposition 2.1 (see [11]) . Let X ∈ M θ and let O X ⊂ M θ be its T M -orbit. Then (1) dim O X = 3 if and only if I X = ∅ . (2) dim O X = 2 if and only if I X is a perfect matching. (3) dim O X = 1 if and only if I X contains a big component which is a cycle.In this case I X is a union of two perfect matchings. (4) dim O X = 0 if and only if I X contains a big component which has twotrivalent vertices of different colors and all other vertices of valence . Inthis case I X is a union of three perfect matchings. For any subset I ⊂ Q , we define a C Q -representation X I = ( X I,a ) a ∈ Q ofdimension α by the rule (for a ∈ Q ) X I,a = ( , a ∈ I, , a I. We say that I is W -compatible if X I is an A -representation. For example, allperfect matchings and an empty set are W -compatible. We say that I is θ -stable if I X is θ -stable. The elements of the fan of M θ are in bijection with W -compatible θ -stable subsets of Q . The rays of the fan of M θ are parametrized by θ -stableperfect matchings. All elements χ I ∈ M ∨ , I ∈ A , are contained in the hyperplane { y ∈ M ∨ Q | ω ∗ M ( y ) = 1 } , where ω M : Z → M was defined earlier. This implies that M θ is a toric 3-Calabi-Yau variety. The above proposition gives an algorithm to construct its toric diagram(this is an intersection of cones of the fan of M θ with the above hyperplane).3. Toric quotients
In this section we will recall some facts from [15] about toric quotients of toricvarieties and give further information on the line bundles on such quotients. Werefer to [6] and [14] for the relevant definitions and properties of toric varieties.Consider a pair (Λ , P ), where Λ is a lattice (i.e. a free abelian group of finiterank) and P ⊂ Λ Q is a polyhedral cone. We associate with it a scheme X P = X (Λ , P ) := Spec C [ P ∩ Λ] . More generally, given a pair (Λ , P ), where Λ is a lattice and P ⊂ Λ Q is apolyhedron, we associate with it a scheme X (Λ , P ) in the following way. Let C ( P ) ⊂ Q × Λ Q be a cone which is a closure of { λ (1 , x ) | λ ∈ Q ≥ , x ∈ P } . We endow C [ C ( P ) ∩ ( Z × Λ)] with a Z -grading induced by the first coordinate anddefine X P = X (Λ , P ) := Proj C [ C ( P ) ∩ ( Z × Λ)] . Let T Λ = Hom Z (Λ , C ∗ ). There is a canonical T Λ -action on X P and a canonical T Λ -linearization of the canonical line bundle O (1) on X P . If P is a cone then REPANT RESOLUTIONS AND BRANE TILINGS 5 C ( P ) = Q ≥ × P and C ( P ) ∩ ( Z × Λ) = N × ( P ∩ Λ). So our new definition of X P is compatible with the old one.The scheme X P can be described as a toric variety associated to a fan. For any y ∈ Λ ∨ Q define face y ( P ) := { x ∈ P | h x, y i = min P h− , y i} . All faces of P have this form for some y ∈ Λ ∨ Q . For any face F ⊂ P , define itsnormal cone N F = N F P := { y ∈ Λ ∨ Q | face y ( P ) ⊃ F } = { y ∈ Λ ∨ Q | h F, y i ≤ h
P, y i} . For any faces
F, G ⊂ P , we have F ⊂ G if and only if N G P ⊂ N F P . The set ofcones N ( P ) = { N F P | F face of P } is a fan in Λ ∨ and the associated toric variety is isomorphic to X P (cf. [15, Prop.2.17]). Lemma 3.1.
Let F ⊂ P be some face and let h F i ⊂ Λ Q be a vector space generatedby the differences x − y , for x, y ∈ F . Then h F i = N ⊥ F .Proof. We can suppose that 0 ∈ F . Then y ∈ N F if and only if h F, y i = 0 , h P, y i ≥ . This implies that h F i ⊂ N ⊥ F . This vetor spaces have equal dimension as dim F +dim N F = dim Λ Q . (cid:3) For any face F ⊂ P , let O F denote the T Λ -orbit corresponding to N F . Lemma 3.2 (see [15, Prop. 2.13])) . Let F ⊂ P be some face. Then (1) dim O F = dim F . (2) The character group of the stabilizer of O F in T Λ equals coker( N ⊥ F ∩ Λ → Λ) . (3) The closure of O F equals X (Λ , F ) . (4) For any two faces
F, G we have F ⊂ G if and only if O F ⊂ O G . Consider an exact sequence of lattices0 → M i −→ Λ d −→ B → . For any face F ⊂ P , we define F M = F ∩ M Q . There is an inclusion T B ⊂ T Λ thatinduces an action of T B on X P and on the line bundle O (1). It is shown in [15,Prop. 3.2] that the corresponding GIT quotient is given by X (Λ , P ) //T B = X ( M, P M ) . Lemma 3.3 ([15, Lemma 3.3]) . Let F ⊂ P be a face. Then (1) O F is T B -semistable if and only if F ∩ M Q = ∅ . (2) O F is T B -stable if and only if M Q intersects inn( F ) transversally. (3) The image of a T B -semistable orbit O F in X P //T B is O F M . Remark 3.4.
Condition that M Q intersects inn( F ) transversally means thatinn( F ) ∩ M Q = ∅ and h F i + M Q = Λ Q . We say that F is M -stable in this case. Wedenote the subscheme of stable points of X P by X sP . Remark 3.5.
There is a bijection between the faces of P M and the faces F ⊂ P such that inn( F ) ∩ M Q = ∅ . MARTIN BENDER AND SERGEY MOZGOVOY
Proposition 3.6.
The set of cones N s ( P M ) = { N F M P M ⊂ M ∨ Q | F ⊂ P is M -stable } forms a fan in M ∨ . The corresponding toric variety is isomorphic to X sP //T B .Proof. The set N s ( P M ) is a subset of the fan N ( P M ). To show that N s ( P M ) is afan, we just have to show that any face of the cone from N s ( P M ) is contained in N s ( P M ). Let F ⊂ P be M -stable. Let τ ′ ⊂ N F M P M be some face. We will showthat τ ′ ∈ N s ( P M ). We can find some face G ′ ⊂ P M such that τ ′ = N G ′ P M . Wechoose a minimal face G ⊂ P such that G M = G ′ . The minimality property impliesthat inn G ∩ M Q = ∅ . Moreover, N G M P M ⊂ N F M P M , so F M ⊂ G M and therefore F ⊂ G . In particular, h G i + M Q = Λ Q and therefore G is M -stable. This impliesthat τ ′ ⊂ N s ( P M ). (cid:3) Lemma 3.7.
Let F ⊂ P be an M -stable face. Consider the normal cones N F = N F P ⊂ Λ ∨ Q and N F M = N F M P M ⊂ M ∨ Q . Then the map i ∗ : Λ ∨ Q → M ∨ Q restricts toa bijection i ∗ F : N F → N F M . Proof.
Without loss of generality we may assume that 0 ∈ inn( F ). Then for any y ∈ N F , we have h F, y i = 0 , h P, y i ≥ . This implies that h F M , i ∗ ( y ) i = 0 , h P M , i ∗ ( y ) i ≥ i ∗ ( y ) ∈ N F M .It follows from our assumption that the vector space h F i intersects M Q transver-sally. This implies that the homomorphism of vector spaces F ⊥ → F ⊥ M is an isomorphism. Therefore the map i ∗ : N F → N F M is injective, as N F ⊂ F ⊥ .Let us prove the surjectivity. Consider y ′ ∈ N F M ⊂ F ⊥ M . We can find y ∈ F ⊥ such that i ∗ ( y ) = y ′ . We know that h P M , y i ≥ h P, y i ≥
0, as this will imply y ∈ N F . Assume that there exists x ∈ P such that h x , y i <
0. Without loss of generality we may assume that Λ Q is generated by x and F , and that P is a convex hull of x and F . Then y ⊥ = h F i . It follows fromthe transversality of the intersection of M Q and F that M Q intersects P \ F . But forany point x in this intersection we have h x, y i < x ∈ P M . This contradictsour assumption h P M , y i ≥ (cid:3) Corollary 3.8.
For any M -stable face F ⊂ P , we have N ∨ F M = N ∨ F ∩ M Q . Proof.
We have N ∨ F ∩ M Q = { x ∈ M Q | h x, N F i ≥ } = { x ∈ M Q | h x, N F M i ≥ } = N ∨ F M . (cid:3) Corollary 3.9.
Let F ⊂ P be an M -stable face. Let σ = N F P , σ ′ = N F M P M , U σ = C [ σ ∨ ∩ Λ] , and U σ ′ = C [( σ ′ ) ∨ ∩ M ] . Then the map X sP → X sP //T B is givenover U σ ′ by U σ = Spec C [ N ∨ F ∩ Λ] → Spec C [ N ∨ F ∩ M ] = U σ ′ . REPANT RESOLUTIONS AND BRANE TILINGS 7
Recall that with any N ( P )-linear support function h : | N ( P ) | → Q (see e.g.[14, Section 2.1]) we can associate a T Λ -equivariant line bundle L h over X P (see[14, Prop. 2.1]). If T B acts freely on X sP then this line bundle descends to a T M -equivariant line bundle on X sP //T B (this follows from [13, Prop. 0.9] and descenttheory). We are going to describe an N s ( P M )-linear support function that givesthis line bundle. Theorem 3.10.
Assume that T B acts freely on X sP . Let h : | N ( P ) | → Q bean N ( P ) -linear support function. Define an N s ( P M ) -linear support function h ′ : | N s ( P M ) | → Q by the rule h ′ ( y ) = h (( i ∗ F ) − ( y )) , y ∈ N F M , where F ⊂ P is an M -stable face and i ∗ F : N F → N F M is a bijection defined earlier.Then the descend of L h to X sP //T B is isomorphic to L h ′ as a T M -equivariant linebundle.Proof. Let us recall the construction of a T Λ -equivariant line bundle L h on X P associated to the support function h : | N ( P ) | → Q (see [14, Prop. 2.1]). For anycommutative semigroup Γ, we denote the canonical basis of the semigroup algebra C [Γ] by ( e γ ) γ ∈ Γ . We can find a system of elements ( l σ ∈ Λ) σ ∈ N ( P ) such that h | σ = l σ | σ for any σ ∈ N ( P ). The line bundle L h is defined by gluing the linebundles U σ × C over U σ , σ ∈ N ( P ) using the gluing functions g τσ : ( U σ × C ) | U τ → U τ × C , ( x, c ) ( x, e l σ − l τ ( x ) c )for τ < σ . The action of T Λ on U σ × C is given by t ( x, c ) = ( tx, e − l σ ( t ) c ) , t ∈ T Λ . Let now σ = N F and σ ′ = N F M , for some M -stable face F ⊂ P . Let π : U σ → U σ ′ = U σ //T B be a canonical projection We give an explicit description of thedescend line bundle ( U σ × C ) //T B over U σ //T B = U σ ′ .The character group of the stabilizer of O F in T B is given by coker( σ ⊥ ∩ Λ → B )(see e.g. [15, Prop. 2.6]). By our assumptions this stabilizer is trivial, so( σ ⊥ ∩ Λ) + M = Λ . This means that we can find some m σ ∈ M such that l σ − m σ ∈ σ ⊥ . Consider theaction of T M on U σ ′ × C given by t ( x, c ) = ( tx, e − m σ ( t ) c ) , t ∈ T M . The map π : U σ × C → U σ ′ × C , ( x, c ) ( π ( x ) , e l σ − m σ ( x ) c ) , is T Λ -equivariant. Indeed, for any t ∈ T Λ we have π ( t ( x, c )) = π ( tx, e − l σ ( t ) c ) = ( π ( x ) , e l σ − m σ ( tx ) e − l σ ( t ) c ) = ( π ( x ) , e − m σ ( t ) e l σ − m σ ( x ) c ) . On the other hand tπ (( x, c )) = t ( π ( x ) , e l σ − m σ ( x ) c ) = ( π ( x ) , e − m σ ( t ) e l σ − m σ ( x ) c ) . This shows that π : U σ × C → U σ ′ × C is a quotient with respect to the action of T B . MARTIN BENDER AND SERGEY MOZGOVOY
The gluing of line bundles ( U σ //T B ) × C is induced by the gluing of line bundles U σ × C and is given by the formula U τ ′ × C → ( U σ ′ × C ) | U τ ′ , ( x, c ) ( x, e m σ − m τ ( x ) c ) , where σ = N F , σ ′ = N F M , τ = N G , τ ′ = N G M for M -stable faces F ⊂ G of P . Thecorresponding support function h ′ : | N ( P M ) | → Q is given on y ′ ∈ σ ′ by h ′ ( y ′ ) = m σ ( y ′ ) = m σ (( i ∗ F ) − ( y ′ )) = l σ (( i ∗ F ) − ( y ′ )) = h (( i ∗ F ) − ( y ′ )) , as l σ − m σ ∈ σ ⊥ . (cid:3) Any element θ ∈ B can be considered as a character θ : T B → C ∗ . We can tensorthe action of T B on O (1) with this character. The stable (resp. semistable) pointsof X P with respect to this linearization are called θ -stable (resp. θ -semistable). Thecorresponding GIT quotient is denoted by X P // θ T B . We have (see [15, 2.16]) X P // θ T B ≃ X (Λ , P θ ) //T B = X ( M, P θ ∩ M Q ) , where P θ = P − λ for some λ ∈ Λ with d ( λ ) = θ .4. Tilting bundles
Let (
Q, W ) be a quiver potential associated to some consistent brane tiling, let A = C Q/ ( ∂W ), and let α = (1 , . . . , ∈ Z Q . The goal of this section is to give atoric description of the tilting bundles on the moduli spaces M θ ( A, α ). Definition 4.1.
Let X be an algebraic variety. A coherent sheaf T ∈ Coh X iscalled a tilting sheaf if Ext n ( T, T ) = 0 for n > D b ( X ) = D b (Coh X ) is generated by the summands of T . A collection of coherentsheaves ( T i ) i ∈ I is called a tilting collection if Ext n ( T i , T j ) = 0 for n > i, j ∈ I ,and the triangulated category D b ( X ) is generated by the objects T i , i ∈ I .We use notation from Section 2. In particular, we have defined an exact sequenceof free abelian groups 0 → M i −→ Λ d −→ B → P ⊂ Λ Q there. Let θ ∈ B be α -generic. We have seen that the modulispace M θ = M θ ( A, α ) is a toric quotient M θ = Spec C [ P ∩ Λ] // θ T B = X P // θ T B = X (Λ , P θ ) //T B = X ( M, P θ ∩ M Q ) , where P θ = P − λ for some λ ∈ Λ with d ( λ ) = θ .We know already how to parametrize the set of T M -orbits of M θ , or equivalently,the fan of M θ . The set of rays of the fan of M θ is in bijection with θ -stable perfectmatchings. It is also in bijection with the the set of facets (codimension 1 faces) of P θM = P θ ∩ M Q .For any weak path u and for any perfect matching I ∈ A , we define χ I ( u ) = χ I (wt( u )). The extremal rays of P ∨ are parametrized by the perfect matchings (see[4, Lemma 2.3.4]). This implies that for any weak path u the system of integers( χ I ( u )) I ∈A determines a T Λ -Cartier divisor and therefore a T Λ -equivariant linebundle over X P which we denote by L ( u ) (forgetting the T Λ -action, we get justa trivial line bundle). If we restrict this system of integers to θ -stable perfectmatchings, we get a T M -Cartier divisor and a T M -equivariant line bundle over M θ which we denote by L ( u ). REPANT RESOLUTIONS AND BRANE TILINGS 9
Let us fix some vertex i ∈ Q . For any vertex i ∈ Q we fix some weak path u i : i → i . The following result proves a conjecture of Hanany, Herzog and Vegh[7, Section 5.2] Theorem 4.2.
For any α -generic θ ∈ B , the line bundles L ( u i ) , i ∈ Q , form atilting collection on M θ ( A, α ) .Proof. We know from [16, Theorem 6.3.1] and the fact that A = C Q/ ( ∂W ) is anon-commutative crepant resolution of its center [11, 3] that there is an equivalenceof categories Ψ : D b (mod A op ) → D b (Coh M θ ) , M M ⊗ LA U , where U is a universal vector bundle on M θ (see also [8]). This implies, in particular,that the vector bundle U = Ψ( A ) is a tilting sheaf. We will give its toric description.Let us recall the construction of the universal vector bundle from [10, Prop. 5.3].Let ( e i ) i ∈ Q be the canonical basis of Z Q and let T := Hom( Z Q , C ∗ ) =GL α ( C ). Let R = R ( A, α ) and let R s ⊂ R be the subvariety of θ -stable repre-sentations. The diagonal ∆ = C ∗ ⊂ T acts trivially on R . We have T B = T / ∆and M θ = R// θ T B = R s /T B .For any i ∈ Q , we define a T -equivariant line bundle L i over R to be R × C with an action on the second factor induced by e i . Explicitly, the action is givenby t ( x, c ) = ( tx, t i c ) , t = ( t i ) i ∈ Q ∈ T , ( x, c ) ∈ R × C . The action of T B on L i is not well defined as ∆ acts nontrivially on L i . Namely, itacts with weight 1 on the second factor. To overcome this problem, we can multiplythe action of T with an action going through some homomorphism T → ∆ suchthat the new action restricted to ∆ is trivial (see [10, Prop. 5.3], note that the T -orbits will not change). The homomorphism ψ : T → ∆ is a character of T ,that is, an element ψ ∈ Z Q . The triviality condition means that ψ · α = −
1. Wemake the choice ψ = e i . Then the action of T B = T / ∆ on L i is given by thecharacter e i − e i ∈ B . Let the T B -equivariant line bundle L i on R descend to theline bundle L i on M θ = R s /T B . It is shown in [10, Prop. 5.3] that ⊕ i ∈ Q L i is auniversal vector bundle on M θ .There is a natural action of T Λ on R . In order to extend it to an action on L i = R × C compatible with an action of T B , we have to choose some λ i ∈ Λ suchthat d ( λ ) = e i − e . We choose λ i = wt( u i ) ∈ Λ. The inverse image of L i withrespect to the natural map X P → R (this is a normalization of some irreduciblecomponent of R , see [11]) is given by L ( u i ). This implies that the descent linebundle of L i with respect to R s → M θ is isomorphic to the descent line bundleof L ( u i ) with respect to X sP → R s → M θ . According to Theorem 3.10, thedescent line bundle of L ( u i ) is L ( u i ). This means that L i ≃ L ( u i ). Therefore U ≃ ⊕ i ∈ Q L ( u i ). (cid:3) Remark 4.3. If u, v : i → j are two weak paths then uv − is a weak cycle. Thisimplies that wt( u ) − wt( v ) ∈ M and therefore L ( u ) and L ( v ) are isomorphic linebundles (see [6, Section 3.4]). If we substitute the vertex i by some vertex i ′ , thenthe line bundles L ( u i ) , i ∈ Q should be tensored with a line bundle L ( u ), where u : i ′ → i is any weak path. This ambiguity corresponds to the ambiguity of theuniversal vector bundle over M θ . The universal vector bundle is defined only upto tensoring with a line bundle. Remark 4.4.
The conjecture of [7, Section 5.2] states actually that the collec-tion L ( u i ), i ∈ Q , is an exceptional collection. But this is certainly false, asHom M θ ( L ( u i ) , L ( u j )) = e j Ae i (see Corollary 4.6) is always nonzero. Remark 4.5.
The collection of line bundles L ( u i ) , i ∈ Q on X P has a propertythat for any α -generic θ ∈ B it descends to a tilting collection on M θ . The existenceof such “globally defined” collection was conjectured by Aspinwall [2]. Corollary 4.6.
For any weak path u : i → j in Q , we have (1) H n ( M θ , L ( u )) = 0 , n > . (2) H ( M θ , L ( u )) = e j Ae i , where A = Q / ( ∂W ) .Proof. Let A = C Q/ ( ∂W ). By the proof of the above theorem, the vector bundle U = ⊕ k ∈ Q L ( u k )can be endowed with a structure of a universal vector bundle. The map Ψ : D b (mod A op ) → D b (Coh M θ ) maps the right A -module e k A to the summand L ( u k )of U . This implies, for n > nA op ( e i A, e j A ) = 0 = Ext n M θ ( L ( u i ) , L ( u j ))= Ext n M θ ( O , L ( u j u − i )) = H n ( M θ , L ( u )) . For the Hom-space we getHom A op ( e i A, e j A ) = e j Ae i = Hom M θ ( L ( u i ) , L ( u j ))= Hom M θ ( O , L ( u j u − i )) = H ( M θ , L ( u )) . (cid:3) Examples
In this section we will consider two examples: the suspended pinch point and thequotient singularity C / ( Z × Z ). In the first example we will study all possiblegeneric stabilities and in the second example we will study only three particularstabilities.5.1. Suspended pinch point.
Here we consider a brane tiling called a suspendedpinch point. The corresponding periodic quiver with a fundamental domain is givenin Figure 1.Let Q = ( Q , Q , Q ) be the corresponding quiver embedded in a torus. We willdenote the arrow from a vertex i ∈ Q to a vertex j ∈ Q by ij . The list of allperfect matchings of the brane tiling is given in Table 1. Every perfect matching isdescribed there as a subset of Q .Recall that we have defined a linear map χ : M → Z A in Section 2. We canchoose such basis of M that the map χ ∗ : Z A → M ∨ is given by the matrix (cid:16) (cid:17) and the map ω ∗ M : M ∨ → Z is given by the matrix ( ). This gives us the toricdiagram of Spec C [ M + ] REPANT RESOLUTIONS AND BRANE TILINGS 11
Figure 1: Periodic quiver and a fundamental domain for SPP
N I ,
312 21 ,
133 32 ,
114 23 ,
115 12 ,
136 21 , I I I I I , I For any α = (1 , , θ ∈ B , the fan Σ θ of M θ has five rays. The matrixof χ ∗ θ : Z Σ θ (1) → M ∨ equals (cid:16) (cid:17) and is independent of the stability θ . ThePicard group Pic( M θ ) is isomorphic to the cokernel of χ θ : M → Z Σ θ (1) (see [6,Section 3.4]). We can choose a basis of Pic( M θ ) such the matrix of Z Σ θ (1) → Pic( M θ ) is given by (cid:0) − −
10 1 − − (cid:1) . There are 6 different chambers of α -generic stabilities. Their representatives aregiven in Table 2.It is easy to see that the perfect matching I is stable with respect to θ , θ ,and θ . The perfect matching I is stable with respect to θ , θ , and θ . Theother perfect matchings are extremal and therefore stable with respect to all θ i , i = 1 , . . . , θ -stable if their union is θ -stable.One can see that the pair { I , I } is stable only with respect to θ and θ . Thepair { I , I } is stable only with respect to θ and θ . This uniquely determines thetriangulation of the toric diagram for any generic stability. N θ − , , , − , , , − , − , − − , , − − , − , α -generic stabilities θ , θ θ , θ θ , θ Figure 2: Triangulations of the toric diagramTo construct the tilting bundle U θ on the moduli space M θ , we choose paths u = e (trivial path), u = 12, u = 13 and intersect them with θ -stable perfectmatchings. This gives us three vectors χ θ ( u i ) ∈ Z Σ θ (1) = Z . Their images withrespect to the map Z Σ θ (1) → Pic( M θ ) (given by the matrix (cid:0) − −
10 1 − − (cid:1) ) deter-mine line bundles which are summands of U θ . The result is given by the followingtable θ χ θ ( u ) χ θ ( u ) χ θ ( u ) U θ θ , θ , θ , , , ,
1) (0 , , , , O ⊕ O (cid:0) − (cid:1) ⊕ O (cid:0) − (cid:1) θ , θ , θ , , , ,
0) (0 , , , , O ⊕ O ( ) ⊕ O ( )5.2. Orbifold C / ( Z × Z ) . Consider a finite abelian group G = Z × Z . Consideran embedding G ⊂ SL ( C ), where the action of the first copy of Z is given by (1 , ,
0) and the action of the second copy of Z is given by (0 , , G can be identified with Z × Z . We will use the following notation for theelements of ˆ G (and sometimes for the elements of G ): a = (1 , b = (0 ,
1) and c = (1 , G is isomorphic to a ⊕ b ⊕ c .We can associate a brane tiling with the above embedding (see [11] for theconstruction and notation). The corresponding periodic quiver with a fundamentaldomain is given in Figure 3.The toric diagram of Spec C [ M + ] is given in Figure 4 (see also [11]). The list ofall possible perfect matchings is given in Table 3.We will consider only stabilities θ = ( − , , , , θ = ( − , − , , , θ = ( − , , , − , where the order of the coordinates of Z Q = Z ˆ G is given by 0 , a, b, c . Note that M θ is isomorphic to Hilb G ( C ).A subset I ⊂ Q is θ -stable if and only if there exists a path in Q \ I from vertex0 ∈ Q to any other vertex of Q . The non-extremal θ -stable perfect matchings are REPANT RESOLUTIONS AND BRANE TILINGS 13 cc0ab a0 aa a acab b bb b b0 0 0000 cc c c c cc 0 0 0c0 c
Figure 3: Periodic quiver for C / ( Z × Z ) b e a e ce Figure 4: Toric diagram for C / ( Z × Z ) N I χ I cb, a, b, ca (2 , , a cb, a, bc, a , , e cb, ab, c , a , , c ac, b , b, ca (0 , , e ac, b , bc, a , , a ac, ab, b, c (0 , , b ba, b , c , ca (0 , , b ba, a, bc, c (2 , , c ba, ab, c , c (0 , , e Table 3: Perfect matchings for C / ( Z × Z ) I , I , I . A subset I ⊂ Q is θ -stable if and only if there exist paths in Q \ I from0 to b and c and from a to b or c . The non-extremal θ -stable perfect matchingsare the same as for θ . A subset I ⊂ Q is θ -stable if and only if there exist pathsin Q \ I from 0 to a , from c to a , and from 0 or c to b . The non-extremal θ -stableperfect matchings are I , I , I .To determine the toric diagram of M θ i , i = 1 , ,
3, we have to find such pairsof θ i -stable perfect matchings that their union is still θ i -stable (we call such pairs θ i -stable). For θ , the pairs { I , I } , { I , I } , { I , I } are stable. The toric diagram of M θ is given in Figure 5. For θ , the pair { I , I } (corresponds to the edge ca )is non-stable. This uniquely determines the toric diagram of M θ (see Figure 5).For θ , the pair { I , I } (corresponds to the edge cb ) is non-stable. This uniquelydetermines the toric diagram of M θ (see Figure 5). e c e ce e e b e a b e a b e ae c Figure 5: Toric diagrams for θ , θ , θ To determine the tilting bundle U θ i on M θ i , i = 1 , ,
3, we choose paths fromvertex 0 ∈ Q to all other vertices of Q and intersect these paths with θ i -stableperfect matchings. We choose paths e , a, b, c . The result of intersecting thesepaths with θ -stable perfect matchings is given in Figure 6. In this way we getCartier divisors for a tilting collection on M θ . The result for θ is the same, as θ -stable perfect matchings and θ -stable perfect matchings coincide. The resultfor θ is given in Figure 7. This gives us Cartier divisors for a tilting collection on M θ . Figure 6: Cartier divisors for a tilting collection on M θ
00 00 00 0 1 001 010 0 0 0 10 0 0 0 1 0
Figure 7: Cartier divisors for a tilting collection on M θ References [1] Klaus Altmann and Lutz Hille,
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