NNON-SIMPLICIAL QUANTUM TORIC VARIETIES
ANTOINE BOIVINA bstract . In this paper, we define quantum toric varieties associatedto an arbitrary fan in a finitely generated subgroup of some R d gener-alizing the article [KLMV20] of Katzarkov, Lupercio, Meersseman andVerjovsky. C ontents
1. Introduction 12. Notations and conventions 32.1. Stacks 32.2. Fans 33. Quantum tori 44. Definition of (non-simplicial) quantum toric varieties 74.1. Affine quantum toric varieties 74.2. Quantum toric varieties 175. GIT-like construction 196. Forgetting calibration and gerbe structure 21References 251. I ntroduction
A toric variety is a complex algebraic variety with an action of an alge-braic torus ( C ∗ ) n with a Zariski open orbit isomorphic to this torus. Sucha variety can be described by a fan of rational strongly convex polyhedralcones on a lattice Γ (see, for example, [CLS11] or [Ful93]).More precisely, there is an equivalence of categories between the cate-gory of toric varieties and the category of fans. This central result gives usa dictionary between geometric properties of toric varieties and combina-torial properties of fans making of toric varieties one of the most studiedclass of complex algebraic varieties.However, in the classical theory, this fan has to be rational. Therefore thetoric varieties are rigid i.e. we can not deform them. Indeed, if we deforma lattice, it can become not discrete (for instance, the group Z + α Z isdiscrete if α is rational but is dense in R if α is irrational)The authors of [KLMV20] gave a construction of "quantum toric vari-ety" (which is a stack) described by a simplicial fan (i.e. the 1-cones of Date : July 1, 2020. a r X i v : . [ m a t h . S G ] J un ANTOINE BOIVIN each cones of the fan are R -linearly independent) on a finitely generated(possibly irrational) subgroup of some R d (named "quantum fan"). Whenthe fan is rational, one recovers the classical toric variety but the caseof irrational simplicial fans is also covered. This construction is functorialand defines an equivalence of categories between the category of quantumtoric varieties and the category of quantum simplicial fans (see theorems5.18 and 6.24 of [KLMV20]).Now, the simplicial fans form only a small part of all the fans. In theclassical theory, they correspond to the toric varieties which are orbifold(i.e. with cyclic singularities). Hence, the restriction to simplicial fans isstrong.In this paper, we extend the construction of [KLMV20] to the generalcase i.e. we define the quantum toric variety associated to an arbitraryfan.The non-simplicial case brings new problems. Indeed, in the simplicialcase, the family of 1-cones of a cone is R -linearly free and can be com-pleted in a basis of R d . Hence, up to isomorphism, a simplicial cone isa standard cone Cone ( e , . . . , e k ) ⊂ R d (where { e , . . . , e d } is the canonicalbasis of R d ). Moreover, we can easily find the faces of these cones: it is thecones generated by a subfamily of { e , . . . , e k } . This basic fact is repeatedlyused in construction of [KLMV20]. It is not the case for non-simplicial caseanymore. Indeed, as the 1-cones of non-simplicial cone is linearly depen-dant, we do not have a notion of standard cone and the cones generatedby a subfamily of the 1-cones is not necessarily a face of the cone since thecone can have an arbitrarily big number of 1-cones.One of the key technical points is to replace the calibration h of thegroup Γ used in [KLMV20] by a calibration ϕ of a group isomorphic to Γ but which is a subgroup of an higher dimensional R p for each maximalcone of the fan ( p is the number of 1-cones of the maximal cone).In section 3, we give a suitable definition of quantum tori in order todeal with the non-simplicial case and more precisely, these quantum toriencode the change of calibration of the last paragraph.In section 4, we define the affine quantum toric variety (which will becalibrated as we have to keep track of the relations between the 1-cones) as-sociated to a (non-simplicial) cone and more generally, the quantum toricvariety with the descent data of the affine pieces associated to a quantumfan. In particular, this construction works for simplicial fans and coincidewith that of [KLMV20].In the end of this section, we prove the main theorem of this paper(extending the similar theorem of [KLMV20]) : Theorem 1.1.
The category of calibrated quantum fans and the category of (cali-brated) quantum toric varieties are equivalent.
Finally, we realize these quantum toric varieties as a global quotient insection 5 and as a gerbe over a "non-calibrated quantum toric varieties"associated to it (i.e. where we replace the action of Z N − d through the ON-SIMPLICIAL QUANTUM TORIC VARIETIES 3 epimorphism ϕ by the action of ϕ ( Z N − d ) and hence removing the ineffec-tivity of the action) in section 6.2. N otations and conventions Stacks.
We will take the same conventions on stacks as [KLMV20] :Let A be the category of affine toric varieties with toric morphisms. Weendow this category with a structure of site with the coverings { U i (cid:44) → T } i ∈{ n } ,where U i are open subsets of T and T = (cid:83) ni = U i .Let G be the category of complex analytic spaces with an action of anabelian complex Lie group G with an open orbit isomorphic to G . Definition 2.1.
Let H be an abelian Lie group and X be an object of G withan action of H commute with the action of G .The stack [ X / H ] is the stack over A whose objects over T are H -principalbundle (cid:101) T → T (in G ) with an H -equivariant morphism (cid:101) T → X and mor-phisms over S → T are a bundle morphism (cid:101) S → (cid:101) T compatible with theequivariant maps.All the stacks in this paper will be of this form or given by the descentdata of stacks of this form.2.2. Fans.
We will recall some definitions on (calibrated) quantum fans :
Definition 2.2.
Let Γ be a finitely generated subgroup of R d such thatVect R ( Γ ) = R d . A calibration of Γ is given by : • A group epimorphism h : Z N → Γ • A subset
I ⊂ {
1, . . . , N } such that Vect C ( h ( e j ) , j / ∈ I ) = C d (this isthe set of virtual generators)This is a standard calibration if Z d ⊂ Γ , h ( e i ) = e i for i =
1, . . . , d and I isof the form { n − |I | +
1, . . . , n } Definition 2.3.
A calibrated quantum fan ( ∆ , h , I ) in Γ is the data of • a collection ∆ of strongly convex polyhedral cones generated byelements of Γ such that every intersection of cones of ∆ is a cone of ∆ , every face of a cone of ∆ is a cone and { } is a cone of ∆ . • a standard calibration h with I its set of virtual generators • A set of generators A i.e. a subset of {
1, . . . , N } \ I such that the1-cone generated by the h ( e i ) for i ∈ A are exactly the 1-cones of ∆ Let h : Z N → Γ a calibration (with I as set of virtual generators) and ( ∆ , h , I ) a calibrated quantum fan in Γ . Note ∆ h the fan in Z N such that(2.1) Cone ( e i , i ∈ I ) ∈ ∆ h if, and only if, Cone ( h ( e i ) , i ∈ I ) ∈ ∆ Definition 2.4.
A morphism of calibrated quantum fans between ( ∆ , h , I ) in Γ and ( ∆ (cid:48) , h (cid:48) , I (cid:48) ) is a pair of linear morphisms ( L : R d → R d (cid:48) , H : R N → R N (cid:48) ) where ANTOINE BOIVIN • The diagram(2.2) R n h R (cid:47) (cid:47) H (cid:15) (cid:15) R dL (cid:15) (cid:15) R n (cid:48) h (cid:48) R (cid:47) (cid:47) R d (cid:48) commutes (where h R (resp. h (cid:48) R ) is the R -linear map associatedto h (resp. to h (cid:48) )) • L ( Γ ) ⊂ Γ (cid:48) • If σ ∈ ∆ then it exists σ (cid:48) ∈ ∆ (cid:48) such that L ( σ ) ⊂ σ (cid:48) (thanks to thefirst point, the same statement is true for H ) • If H ( e i ) ∈ Cone ( e j , j ∈ I ) ∈ ∆ h then H ( e i ) ∈ (cid:76) j ∈I (cid:48) N e j (samestatement is true for L and the vectors h ( e i ) , thanks to the firstpoint) • For all i / ∈ I , H ( e i ) ∈ (cid:76) j / ∈I (cid:48) Z e j • There exists a map s : I → I (cid:48) such that for all i ∈ I , H ( e i ) = e s ( i ) Remark . We can define non-calibrated counterpart of these definitionsby replacing the data of the calibration by the data of the 1-cones h ( e i ) , i ∈ {
1, . . . , N }
3. Q uantum tori
Let Γ a finitely generated subgroup of R d and h : Z N → Γ a calibrationof Γ (note I the set of virtual generators).The standard calibrated quantum torus associated to these data is the quo-tient stack (cid:84) calh , I = [ C d / Z N ] where Z N acts on C d through the morphism h .However, since a non-simplicial cone can have more than d generators, wehave to consider quantum tori as a quotient stack of a higher-dimensionalspace C p . In order to do this, we will replace the group Γ and its calibra-tion h by a subgroup G of R p isomorphic to it with a calibration ϕ of it.Then, we describe these quantum tori by the data of the underlying stan-dard quantum torus (in order to keep track of the combinatorial data ofthe group Γ ) with the data of a stack isomorphism, respecting the virtualgenerators, with such quotient stack. More precisely, Definition 3.1.
A presented calibrated quantum torus is a 6-uple ( (cid:84) calh , I , ϕ : Z N → G ⊂ C p , I (cid:48) , L , H , s ) where ϕ is a calibration of the group G (with I (cid:48) its set of virtual generators), L : R p → R d is a linear epimorphism, H : Z N → Z N is a group isomorphism and s : I → I (cid:48) is a bijection suchthat : • L ( G ) = Γ and L | G : G → Γ is a group isomorphism. • The diagram Z N H (cid:47) (cid:47) ϕ (cid:15) (cid:15) Z Nh (cid:15) (cid:15) G L | G (cid:47) (cid:47) Γ is commutative. • For all i ∈ I , H ( e i ) = e s ( i ) and for all i / ∈ I , H ( e i ) ∈ (cid:76) j / ∈I (cid:48) Z e j ON-SIMPLICIAL QUANTUM TORIC VARIETIES 5
The morphism ϕ is the calibration of this presented calibrated quantumtorus.With these data, we can define the quotient stack [ C p / Z N × ker ( L ⊗ R id C )] , where Z N × ker ( L ⊗ R id C ) acts on C p through ( m , w ) · z = z + ϕ ( m ) + w ,Moreover, these data encode a stack isomorphism [ C p / Z N × ker ( L ⊗ R id C )] (cid:39) (cid:84) calh , I .If L is a linear isomorphism then this stack isomorphism is a calibratedtorus isomorphism as defined in [KLMV20].We can define in the same way non-calibrated quantum tori : Definition 3.2.
A presented non-calibrated quantum torus is a couple ( (cid:84) d , Γ , L ) where L is a linear epimorphism R p → R d In particular, the linear morphism descends to a stack isomorphism [ C p / L − C ( Γ )] (cid:39) (cid:84) d , Γ (where L C = L ⊗ R id C and L − C ( Γ ) acts on C p by translations) which is aquantum torus isomorphism (in the sense of [KLMV20]) if L is a linearisomorphism.In both cases, we can define a multiplicative form of the torus thanks tothe exponential map E : ( z , . . . , z p ) ∈ C p (cid:55)→ ( e i π z , . . . , e i π z p ) ∈ T p : = ( C ∗ ) p (1) In the non-calibrated case, we have an isomorphism (see diagram3.12 of [KLMV20]) : [ C p / L − C ( Γ )] (cid:39) [ T p / E ( L − C ( Γ ))] where E ( L − C ( Γ )) acts multiplicatively on T p .(2) In the calibrated case, we have to suppose h standard i.e. we sup-pose that there is a subset (cid:101) I = { i , . . . , i d } ⊂ {
1, . . . , p } such that Z (cid:101) I ⊂ G and h ( e k ) = e i k for k = d . Then, the exponential mapgives us : [ C p / Z N × ker ( L C )] (cid:39) [ T p / Z N − d × E ( ker ( L C ))] where Z N − d × E ( ker ( L C )) acts on T p through ( m , E ( w )) · z = E ( ϕ ( ⊕ m ) + w ) z Definition 3.3.
A morphism of presented non-calibrated quantum tori ( (cid:84) d , Γ , L ) → ( (cid:84) d (cid:48) , Γ (cid:48) , L (cid:48) ) ANTOINE BOIVIN or presented torus morphism is a couple ( L , L (cid:48) ) of linear morphisms suchthat C d L (cid:15) (cid:15) C p / ker ( L C ) [ L C ] (cid:111) (cid:111) L (cid:48) (cid:15) (cid:15) C d (cid:48) C p (cid:48) / ker ( L (cid:48) C ) [ L (cid:48) C ] (cid:111) (cid:111) where [ L ] is the morphism induced by L on the quotient. Definition 3.4.
A morphism of presented calibrated quantum tori ( (cid:84) calh , I , ϕ : Z N → G ⊂ C p , (cid:101) I , L , H , s ) → ( (cid:84) calh (cid:48) , I (cid:48) , ϕ (cid:48) : Z N (cid:48) → G (cid:48) ⊂ C p (cid:48) , (cid:101) I (cid:48) , L (cid:48) , ϕ , s (cid:48) ) or presented calibrated torus morphism is a 6-uple ( L , H , S , L (cid:48) , H (cid:48) , S (cid:48) ) ofmorphisms such that • ( L , H , S ) induces a torus morphism (cid:84) calh , J → (cid:84) calh (cid:48) , J (cid:48) as defined in[KLMV20] (definition 3.11), • for all j ∈ (cid:101) I , H (cid:48) ( e j ) = e S (cid:48) ( j ) and for all j / ∈ (cid:101) I , H (cid:48) ( e j ) ∈ (cid:77) k / ∈ (cid:101) I (cid:48) Z e k • The following diagrams commute : C N H (cid:48) (cid:15) (cid:15) C NH (cid:111) (cid:111) H (cid:15) (cid:15) h (cid:47) (cid:47) C d L (cid:15) (cid:15) C p / ker ( L C ) [ L C ] (cid:111) (cid:111) L (cid:48) (cid:15) (cid:15) C N (cid:48) C N (cid:48) H (cid:48) (cid:111) (cid:111) h (cid:48) (cid:47) (cid:47) C d (cid:48) C p (cid:48) / ker ( L (cid:48) C ) [ L (cid:48) C ] (cid:111) (cid:111) I s (cid:47) (cid:47) S (cid:15) (cid:15) (cid:101) I S (cid:48) (cid:15) (cid:15) I (cid:48) s (cid:48) (cid:47) (cid:47) (cid:101) I (cid:48) In particular, we have the following commutative diagram showing therelation between the different calibrations : Z N ϕ (cid:15) (cid:15) H (cid:47) (cid:47) Z N H (cid:47) (cid:47) h (cid:15) (cid:15) Z N (cid:48) h (cid:48) (cid:15) (cid:15) H (cid:48)− (cid:47) (cid:47) Z N (cid:48) ϕ (cid:48) (cid:15) (cid:15) G L (cid:47) (cid:47) Γ L (cid:47) (cid:47) Γ (cid:48) ( L (cid:48)| G (cid:48) ) − (cid:47) (cid:47) G (cid:48) We can reformulate this in term of quotient stack :
Lemma 3.5.
We use the same notations as definition 3.4.Let (cid:108) (cid:48) : [ C p / Z N × ker ( L C )] → [ C p (cid:48) / Z N (cid:48) × ker ( L (cid:48) C )] be the stack morphismdescribed by the linear morphisms ( L (cid:48) , H (cid:48) ) and (cid:108) cal : (cid:84) cald , Γ → (cid:84) cald (cid:48) , Γ (cid:48) be the stackmorphisms described by the linear morphisms ( L , H ) . Then, the diagram ON-SIMPLICIAL QUANTUM TORIC VARIETIES 7 [ C p / Z N × ker ( L C )] (cid:108) (cid:48) (cid:47) (cid:47) (cid:39) (cid:15) (cid:15) [ C p (cid:48) / Z N (cid:48) × ker ( L (cid:48) C )] (cid:39) (cid:15) (cid:15) (cid:84) cald , Γ (cid:108) cal (cid:47) (cid:47) (cid:84) cald (cid:48) , Γ (cid:48) commutes (where the isomorphisms are the stack isomorphisms encoded by thepresentation of the presented calibrated quantum tori). The presented quantum tori are (essentially) not new quantum tori. Infact, all presented quantum tori with the same underlying quantum toriare isomorphic :
Lemma 3.6. ( (cid:84) calh , I , ϕ : Z N → G ⊂ C p , I (cid:48) , L , H , s ) and ( (cid:84) calh , I , h : Z N → Γ , I , id , id , id ) (resp. ( (cid:84) d , Γ , L ) and ( (cid:84) d , Γ , id ) ) are isomorphic.Proof. An isomorphism is given by ( id , id , id , [ L ] , H , s ) (resp. ( id , [ L ]) ). (cid:3) Proposition 3.7.
The forgetful functor ( (cid:84) calh , I , ϕ : Z N → G , I (cid:48) , L , H , s ) → (cid:84) calh , I , resp. ( (cid:84) d , Γ , L ) → (cid:84) d , Γ , is an equivalence of categories between the category of pre-sented calibrated quantum tori and the category of standard calibrated quantumtori, resp. between the category of non-calibrated quantum tori and the categoryof non-calibrated standard quantum tori.Proof. Use lemma 3.6 (cid:3)
4. D efinition of ( non - simplicial ) quantum toric varieties Let Γ be a finitely generated subgroup of R d such that Vect R ( Γ ) = R d , ( ∆ , h cal : Z N → Γ , I ) a standard calibrated quantum fan (with I its set ofvirtual generators). In order to define a quantum variety associated to thisfan, we will describe the affine quantum variety associated to each cone ofthis fan. The crucial point is to replace the calibration h of Γ by an adaptedcalibration ϕ for each maximal cone σ of the fan ∆ .After that, we will define the quantum toric varieties with the descentdata of affine pieces as [KLMV20].At the end of this section, we will prove that the correspondence ( ∆ , h cal : Z N → Γ , I ) (cid:55)→ (cid:88) cal ∆ , h cal , I is an equivalence of categories.4.1. Affine quantum toric varieties.
Let σ = σ I = Cone ( v i : = h cal ( e i ) , . . . , v i p : = h cal ( e i p )) (with I : = { i , . . . , i p } ⊂ {
1, . . . , N } ) be a strongly convex cone of ∆ . Thereare two possibilities : σ is of maximal dimension i.e. of dimension d ornot. In the latter, we have to do a choice of a completion of Vect R ( σ ) in ANTOINE BOIVIN R d . Fortunately, all obtained quantum varieties are isomorphic and thisisomorphism respects a cocycle condition.4.1.1. Cones of dimension d.
Suppose σ is a cone of dimension d i.e. Vect R ( σ ) = R d .Note h σ : Z I → Γ the restriction of h cal on Z I , h σ C : C I → C d the C -linearmap associated to it and (cid:98) σ the cone of R N defined by(4.1) (cid:98) σ = Cone ( e i , i ∈ I ) and (cid:101) σ its restriction on R I Let B = ( v i , i ∈ (cid:101) I ) a subfamily of ( v , . . . , v p ) which is a basis of C d .Then, we will consider the decomposition C I = C (cid:101) I ⊕ ker ( h σ C ) . The map h σ C induces a linear isomorphism ψ between C (cid:101) I and C d .Let χ ∈ S N be a permutation such that χ ( {
1, . . . , d } ) = (cid:101) I . Definition 4.1.
The linear morphism ϕ : C N → C (cid:101) I (cid:44) → C I defined by e k (cid:55)→ ψ − ( h cal C ( e χ ( k ) )) is called calibration associated to σ (and B and χ )We can think this morphism as a calibration induced by the calibra-tion h on C I . Indeed, the image of Z d ⊕ ϕ is Z (cid:101) I (byconstruction) and the group ϕ ( Z N ) is isomorphic to Γ .We can define an action of Z N − d × E ( ker ( h σ C )) on C I by setting for ( m , E ( t )) ∈ Z N − d × E ( ker ( h σ C )) and z ∈ C I , ( m , t ) · z = E ( ϕ ( ⊕ m )) + t ) zRemark . The non-effectiveness of this action only comes from the cali-bration i.e. the subgroup of ineffectivity of this action is ker ( h cal ) .The action of T I on C I = U (cid:101) σ commutes with this action. Hence, we canform the quotient [ C I / Z N − d × E ( ker ( h σ C ))] . Lemma 4.3.
The quotient stack [ C I / Z N − d × E ( ker ( h σ C ))] depends neither onthe choice of the basis B nor on the permutation χ . Hence, the action induced bythe morphism ϕ depends only on σ .Proof. Let ( B = ( v i , i ∈ (cid:101) I ) , χ ) and ( B (cid:48) = ( v i , i ∈ (cid:101) I (cid:48) ) , χ (cid:48) ) be two pairs ofbasis and permutation. Then we can define two associated isomorphisms ψ : C (cid:101) I → C d and ψ (cid:48) : C (cid:101) I (cid:48) → C d and two morphisms ϕ : C N → C I , ϕ (cid:48) : C N → C I (cid:48) . Then, the identity is a α -equivariant morphism, where α is the morphism Z N − d × E ( ker ( h σ )) → Z N − d × E ( ker ( h σ )) defined by α ( m , E ( y )) = ( P − χ (cid:48) P χ ( m ) , E ( ϕ ( m ) − ψ (cid:48)− h ( ϕ ( m ))) E ( y )) where P χ is the linear map associated to χ .Indeed, the following diagram is commutative ON-SIMPLICIAL QUANTUM TORIC VARIETIES 9 E ( h − σ ( Γ )) = (cid:47) (cid:47) E ( h − σ ( Γ )) Z N − d × E ( ker ( h σ )) action through ϕ (cid:79) (cid:79) α (cid:47) (cid:47) Z N − d × E ( ker ( h σ )) action through ϕ (cid:48) (cid:79) (cid:79) (cid:3) Definition 4.4.
The stack (cid:85) cal σ : = [ C I / Z N − d × E ( ker ( h σ C ))] is the quantumtoric variety associated to the cone σ and to the calibration h cal : Z N → Γ . Remark . If σ is non-simplicial (i.e. ker ( h σ C ) is non-empty and thusis not discrete) then the quantum toric variety associated to σ cannot bedescribe as a quasifold (i.e. locally the quotient of a space R n by a discretesubgroup, see [Pra01]) in contrast to the simplicial case.The associated "torus" to (cid:85) cal σ is the stack [ T I / Z N − d × E ( ker ( h σ C ))] . Wecan endow this stack with a structure of presented calibrated quantumtorus thanks to the morphisms used to define (cid:85) cal σ : Proposition 4.6. ( (cid:84) calh , I , ϕ : Z N → ψ − ( Γ ) , h σ C , χ − ( I ) , P χ : e i (cid:55)→ e χ ( i ) , χ ) is a presented calibrated quantum torus which encodes the stack [ T I / Z N − d × E ( ker ( h σ C ))] Proof.
This proposition comes from the fact that ψ is induced by h σ C on C I / ker ( h σ C ) (cid:39) C (cid:101) I and the equality ϕ = ψ − hP χ . (cid:3) Notation 4.7.
In what follows, we will omit the isomorphism and justwrite [ T p / Z N − d × E ( ker ( h σ C ))] instead of this 6-uple. Warning 4.8.
In the classical non-simplicial case (i.e. Γ is a lattice and thecalibration is an isomorphism), the stack (cid:85) cal σ is not a (stack representable by a)variety. However, if we replace the stack quotient by the GIT quotient, we getthe classical toric variety associated to σ : Proposition 4.9. If Γ is discrete, h is an isomorphism and the set of virtualgenerators is empty, then we can define the (classical) toric variety U σ associatedto σ (and Γ ) and U σ = C I (cid:12) Z N − d × E ( ker ( h σ C )) where the (cid:12) denotes the GIT quotient. Moreover, all the points of C I are semi-stable (see the definition in [MFK94] ) for the action of Z N − d × E ( ker ( h σ C )) for the trivial line bundle C I × C → C I with the linearization defined for all ( m , t ) ∈ Z N − d × E ( ker ( h σ C )) and for ( z , w ) ∈ C I × C , ( m , t ) · ( z , w ) = (( m , t ) · z , E ( (cid:104) ϕ ( ⊕ m ) + t , a (cid:105) ) w ) for any a ∈ Z I .Proof. Since h is an isomorphism, the action of the group Z n − d × E ( ker ( h σ C )) on C I is the same action as the action of E ( h − σ C ( Γ )) on it thanks to theequality(4.2) { E ( ϕ ( ⊕ m ) + t ) | m ∈ Z N − d , E ( t ) ∈ E ( ker ( h σ C )) } = E ( h − σ C ( Γ )) . If we suppose Γ discrete, we can suppose too that Γ = Z d without lossof generality. In [CLS11] (theorem 5.1.11), the authors prove that C I → U σ is an almost geometric quotient for the action of G defined by G : = Hom Z ( coker ( M = ( v i , i ∈ I ) T : Z I → Z d ) , C ∗ ) (The cokernel in this expression is also defined as the class group of thevariety U σ see [CLS11] p172/173).Since C ∗ is a divisible group, we have a exact sequence :(4.3) 0 (cid:47) (cid:47) G (cid:47) (cid:47) T I ϕ (cid:47) (cid:47) T d (cid:47) (cid:47) ϕ is the induced morphism by Hom ( M , C ∗ ) and theisomorphisms Hom ( Z I , C ∗ ) → T I , ( u : Z I → C ∗ ) (cid:55)→ ( u ( e i ) , i ∈ I ) andHom ( Z d , C ∗ ) → T d , ( u : Z d → C ∗ ) (cid:55)→ ( u ( e ) , . . . , u ( e d )) .Then, we deduce from the equality h σ C = M T C that ϕ is the morphism T I → T d induced by h . Finally, thanks to the exact sequence (4.3), we getthe isomorphism G (cid:39) E ( h − σ C ( Z d )) (since ker ( E ) = Z d ) The semi-stability is proved in chapter 12 of [Dol03]. (cid:3)
We cannot have the equality C I (cid:12) Z N − d × E ( ker ( h σ C )) = [ C I / Z N − d × E ( ker ( h σ C ))] .Indeed, the categorical quotient (for the action of G ) C I → U σ is geometricif, and only if, σ is a simplicial cone because the set of semi-stable pointsis equal to the set of stable points if, and only if, the cone is simplicial (see[Dol03] proposition 12.1) Example 4.10.
Let Γ = Z e + Z e + Z e + Z ( ae − be + ce ) + N ∑ i = Z v i ⊂ R with a , b , c ∈ R > , v i ∈ R be a subgroup of R , h cal : Z N → Γ be thecalibration of Γ defined by h cal ( e i ) = e i for i =
1, 2, 3, h cal ( e ) = ae − be + ce and h cal ( e i ) = v i for i ≥
5. Let σ = Cone ( e , e , e , ae − be + ce ) be astrongly convex cone of R .The morphism h σ is the restricted map h cal | Z ⊕ and the kernel ker ( h σ C ) is the line ker ( h σ C ) = C ( − a , b , − c , 1 ) We have a decomposition C = ( C ⊕ ) ⊕ ker ( h σ C ) and hence, an actionof Z N − × E ( ker ( h σ C )) on C defined by :(4.4) ( m , E ( t )) · s = E ( h ( ⊕ m ) + t ) s Then, (cid:85) cal σ = [ C / Z N − × E ( C ( − a , b , − c , 1 ))] For the case N = a = b = c =
1, thanks to the decomposition of (cid:83) , we find a stacky version of the toric variety C (cid:12) G = V ( yt − xz ) ⊂ C (see [Ful93] p17) ON-SIMPLICIAL QUANTUM TORIC VARIETIES 11 F igure
1. The cone σ Cone of dimension k < d. Suppose σ = σ I ⊂ R d is a cone of dimen-sion k < d .Let J a subset of [[ N ]] of cardinal d − k such that C d = Vect ( σ ) ⊕ Vect ( v j , j ∈ J ) Hence, h σ : C I ⊕ C J → C d , e i (cid:55)→ v i is an epimorphism. Now, we canreuse the discussion of subsection 4.1.1 and define an action of Z N − d × E ( ker ( h σ C )) on C I ⊕ C J (we will suppose that the permutation χ sends { k +
1, . . . d } on J ). We remark that the toric variety C I × T J = U (cid:101) σ × ispreserved by this action and thus, we can define the affine quantum toricvariety associated to σ : (cid:85) cal σ : = [ C I × T J / Z N − d × ker ( h σ )] Proposition 4.11.
The affine quantum toric variety (cid:85) cal σ is well-defined i.e. ifwe change the family ( v j , j ∈ J ) , we get an isomorphism which respects a cocyclecondition.Proof. Let J and J (cid:48) be two subsets of [[ N ]] of cardinal d − k such thatdim ( Vect ( v j , j ∈ J )) = dim ( Vect ( v j , j (cid:48) ∈ J (cid:48) )) = d − k and χ , χ (cid:48) be theassociated permutations (and P χ , P χ (cid:48) the associated linear maps). Themorphism P χ (cid:48) ◦ P − χ : C J → C J (cid:48) is an isomorphism and id ⊕ P χ (cid:48) ◦ P − χ : C I ⊕ C J → C I ⊕ C J (cid:48) too.We will prove that this morphism induces an isomorphism of stacks [ C I × T J / Z N − d × ker ( h σ C ⊕ h J )] (cid:39) [ C I × T J (cid:48) / Z N − d × ker ( h σ C ⊕ h J (cid:48) )] where h J (resp. h J (cid:48) ) is the restriction of h cal C on Z J (resp. Z J (cid:48) )Firstly, we can remark that x ⊕ y ∈ ker ( h σ C ⊕ h J ) = ker ( h σ C ⊕ h J (cid:48) ) if, andonly if, x ∈ ker ( h σ C ) and y =
0. Thus, we get the following commutativediagram (every arrow is an isomorphism) : C I ⊕ C J / ker ( h σ C ⊕ h J ) id ⊕ P χ (cid:48) ◦ P − χ (cid:47) (cid:47) (cid:15) (cid:15) C I ⊕ C J (cid:48) / ker ( h σ C ⊕ h J (cid:48) ) (cid:15) (cid:15) C I / ker ( h σ C ) ⊕ C J (cid:47) (cid:47) C I / ker ( h σ C ) ⊕ C J (cid:48) By construction, the isomorphism id ⊕ P − χ (cid:48) ◦ P χ descends to quotient.Hence, we get : C I ⊕ C J E (cid:47) (cid:47) id ⊕ P χ (cid:48) ◦ P − χ (cid:15) (cid:15) T I × T J (cid:31) (cid:127) (cid:47) (cid:47) (cid:15) (cid:15) C I × T J (cid:47) (cid:47) (cid:15) (cid:15) [ C I × T J / Z N − d × ker ( h σ C ⊕ h J )] (cid:104) JJ (cid:48) (cid:15) (cid:15) C I ⊕ C J (cid:48) E (cid:47) (cid:47) T I × T J (cid:48) (cid:31) (cid:127) (cid:47) (cid:47) C I × T J (cid:47) (cid:47) [ C I × T J / Z N − d × ker ( h σ C ⊕ h J (cid:48) )] The rightmost morphism is the desired morphism, which is an isomor-phism.If we take a third subset J (cid:48)(cid:48) (and a permutation χ (cid:48)(cid:48) ) , we get two otherisomorphisms (cid:104) JJ (cid:48)(cid:48) and (cid:104) J (cid:48) J (cid:48)(cid:48) . The equality ( id ⊕ P χ (cid:48)(cid:48) ◦ P − χ (cid:48) ) ◦ ( id ⊕ P χ (cid:48) ◦ P − χ ) = id ⊕ P χ (cid:48)(cid:48) ◦ P − χ induces an equality between the stack (iso)morphisms (cid:104) J (cid:48)(cid:48) J (cid:48) ◦ (cid:104) J (cid:48) J = (cid:104) J (cid:48)(cid:48) J (cid:3) The following proposition is proved by the same manner as proposition4.9.
Proposition 4.12. If Γ is discrete, h is an isomorphism and the set of virtualgenerators is empty thenU σ = C I × T J (cid:12) Z N − d × E (cid:16) ker (cid:16) h σ (cid:17)(cid:17) Example 4.13.
We will describe a variant of example 4.10 :Let Γ = Z e + Z e + Z e + Z ( ae − be + ce ) + N ∑ i = v i Z ⊂ R be a subgroup of R , h : Z N → Γ be the calibration of Γ defined by h ( e i ) = e i for i =
1, 2, 3, h ( e ) = ae − be + ce and h ( e i ) = v i for i ≥
5, and σ = Cone ( e , e , e , ae − be + ce ) .Note k ∈ {
5, . . . , N } an integer such that ( e , e , e , v k ) is a basis of C .The kernel of the morphism h σ : C ⊕ C { k } → C is C ( − a , b , − c , 1, 0 ) i.e.ker ( h σ C ) ⊕
0. Then, (cid:85) cal σ = [ C × C ∗ / Z N − × E ( C ( − a , b , − c , 1, 0 ))] We will conclude this subsection with the compatibility of the construc-tion with the restriction to a face of a cone.
ON-SIMPLICIAL QUANTUM TORIC VARIETIES 13
Proposition 4.14.
Let σ = σ I be a cone and let τ = σ I (cid:48) be a face of σ . Then wehave an isomorphism (cid:85) cal τ (cid:39) [ C I (cid:48) × T I \ I (cid:48) × T J / Z N − d × E ( ker ( h σ C ))] (cid:44) → (cid:85) cal σ which restricts to a torus isomorphism [ T I (cid:48) × T J (cid:48) / Z N − d × E ( ker ( h τ C ))] (cid:39) [ T I (cid:48) × T I \ I (cid:48) × T J / Z N − d × E ( ker ( h σ C ))] induced by the identity of (cid:84) calh , I .Proof. Without loss of generality, we can suppose that J (cid:48) = J (cid:228) K and C I ∪ J = C (cid:101) I ⊕ ker ( h σ C ) ⊕ C K .Then, the map f : C I (cid:48) × T J (cid:48) = ( C (cid:101) I ⊕ ker ( h τ C )) × T J (cid:48) → ( C (cid:101) I ⊕ E I \ I ( ker ( h σ C )) × T J (cid:48) = C I (cid:48) × T I \ I (cid:48) × T J (where E I \ I : ( z , w ) ∈ C I (cid:48) × C I \ I (cid:48) (cid:55)→ ( z , E ( w )) ∈ C I (cid:48) × T I \ I (cid:48) ) defined by f ( x ⊕ t , y ) = ( x , E I \ I ( t , 0 ) , z ) descends to the desired stack isomorphism (cid:3) Toric morphisms of affine quantum toric varieties.
We will use the samedefinition of toric morphism as [KLMV20] :
Definition 4.15.
A toric morphism between the two affine quantum toricvarieties (cid:85) cal σ = [ C I × T J / G ] and (cid:85) cal σ = [ C I (cid:48) × T J (cid:48) / G ] is a stack mor-phism [ C I × T J / G ] → [ C I (cid:48) × T J (cid:48) / G ] which restricts to a presented cali-brated torus morphism [ T I × T J / G ] → [ T I (cid:48) × T J (cid:48) / G ] By definition of torus morphism, such torus morphism induces a torusmorphism between standard quantum tori (cid:84) calh , I → (cid:84) calh (cid:48) , I (cid:48) and thus two lin-ear maps ( L : R d → R d (cid:48) , H : R N → R N (cid:48) ) . Moreover, L ( σ ) ⊂ σ (cid:48) and H ( (cid:98) σ ) ⊂ (cid:98) σ (cid:48) . These morphisms form a calibrated quantum fan morphism.Conversely, we will discuss how we can associate to a morphism of cali-brated quantum fans a toric morphism :Let σ be a cone of dimension k of R d , σ (cid:48) be a cone of dimension k (cid:48) of R d (cid:48) . Let (cid:101) σ and (cid:101) σ (cid:48) be the associated cone in R N and R N (cid:48) .Let ( L , H ) : ( σ , h ) → ( σ (cid:48) , h (cid:48) ) be a calibrated quantum fan morphism.Let σ = σ I be a cone of dimension k of ∆ and (cid:101) σ the associated cone. Bydefinition of calibrated quantum fan morphism, there exists a cone (cid:101) σ (cid:48) = (cid:102) σ I (cid:48) of ∆ (cid:48) h (cid:48) such that H ( (cid:101) σ ) ⊂ (cid:101) σ (cid:48) .We will adapt the construction of [KLMV20] (section 5.1). Firstly, we beginby replacing the calibration h by the morphism ϕ (used in definition of (cid:85) cal σ ) in diagram (2.2) :Let J be a subset of cardinal d − k of {
1, . . . , N } such that C d = Vect C ( σ ) ⊕ Vect C ( v j , j ∈ J ) .Let (cid:101) J be a subset of J such that for all j ∈ (cid:101) J , L ( v j ) / ∈ L ( Vect C ( σ )) andsuch that the family ( L ( v j ) , j ∈ (cid:101) J ) is free. Let J (cid:48) be a subset of cardinal d (cid:48) − dim ( σ (cid:48) ) of {
1, . . . , N } containing (cid:101) J such that C d (cid:48) = Vect C ( σ (cid:48) ) ⊕ Vect C ( v (cid:48) j , j ∈ J (cid:48) ) .Note h J : C J → C d (resp. h J (cid:48) : C J (cid:48) → C d (cid:48) ) the linear map e j (cid:55)→ v j forall j ∈ J (resp. e j (cid:55)→ v (cid:48) j for all j ∈ J (cid:48) ) To sum up, we have the followingcommutative diagram :(4.5) C I ⊕ C J h σ C + h J (cid:47) (cid:47) (cid:101) L (cid:15) (cid:15) C dL (cid:15) (cid:15) C I (cid:48) ⊕ C J (cid:48) h σ (cid:48) C + h J (cid:48) (cid:47) (cid:47) C d (cid:48) where (cid:101) L is the map C I ⊕ C J = ker ( h σ ) ⊕ ( C (cid:101) I ⊕ C J ) → ker ( h (cid:48) σ (cid:48) ) ⊕ ( C (cid:101) I (cid:48) ⊕ C J (cid:48) ) defined by : (cid:101) L ( w , z ) = ( H ( w ) , ψ (cid:48)− ( L ( h σ C ( z )))) where ψ (cid:48) is the map induced by h (cid:48) σ (cid:48) used in the definition of (cid:85) cal σ . Thismap is well defined i.e. H ( ker ( h σ C )) ⊂ ker ( h (cid:48) σ (cid:48) C ) since H ( (cid:98) σ ) ⊂ (cid:98) σ (cid:48) and ( L , H ) is a morphism of calibrated quantum fans.Let χ and χ (cid:48) be the permutations used to defined the toric varieties (cid:85) cal σ and (cid:85) cal σ (cid:48) i.e. permutations such that χ ( {
1, . . . , k } ) = (cid:101) I , χ ( { k +
1, . . . , d } ) = J and χ (cid:48) ( {
1, . . . , k (cid:48) } ) = (cid:101) I (cid:48) , χ (cid:48) ( { k (cid:48) +
1, . . . , d (cid:48) } ) = J (cid:48) .Let P χ ∈ GL N ( R ) and P χ (cid:48) ∈ GL N (cid:48) ( R ) be the associated linear maps.Moreover, we can extend this diagram with the morphisms ϕ = ψ − ◦ h C ◦ P χ : C N → C I ⊕ C J and ϕ (cid:48) = ψ (cid:48)− ◦ h (cid:48) C ◦ P χ (cid:48) : C N (cid:48) → C I (cid:48) ⊕ C J (cid:48) used todefine the quantum toric varieties (cid:85) cal σ and (cid:85) cal σ (cid:48) (and more precisely, usedto define an action of Z N − d (resp. Z n (cid:48) − d (cid:48) ) on C I ⊕ C J (resp. C I (cid:48) ⊕ C J (cid:48) ) :(4.6) C N ϕ (cid:47) (cid:47) P − χ (cid:48) HP χ (cid:15) (cid:15) C I ⊕ C J h σ C + h J (cid:47) (cid:47) (cid:101) L (cid:15) (cid:15) C dL (cid:15) (cid:15) C N (cid:48) ϕ (cid:47) (cid:47) C I (cid:48) ⊕ C J (cid:48) h σ (cid:48) C + h J (cid:48) (cid:47) (cid:47) C d (cid:48) After this replacement, we can follow the construction of corollary 5.6 of[KLMV20] (i.e. the section 5.1) in order to associate to the morphisms ( H χχ (cid:48) : = P − χ (cid:48) HP χ , (cid:101) L ) a toric morphism (cid:85) cal σ → (cid:85) cal σ (cid:48) :Note E I : C I × C J → T I × C J the map defined by : (( z i ) i ∈ I , ( w j ) j ∈ J ) (cid:55)→ (( E ( z i )) i ∈ I , ( w j ) j ∈ J ) and E J : C I × C J → C I × T J the map defined by : (( z i ) i ∈ I , ( w j ) j ∈ J ) (cid:55)→ (( z i ) i ∈ I , ( E ( w j )) j ∈ J ) ON-SIMPLICIAL QUANTUM TORIC VARIETIES 15
In the same way, we define E (cid:48) I (cid:48) , E (cid:48) J (cid:48) and E (cid:48) = E (cid:48) I (cid:48) ◦ E (cid:48) J (cid:48) . Then, we have thecommutative diagram :(4.7) C I ⊕ C J E I (cid:47) (cid:47) (cid:101) L (cid:15) (cid:15) T I × C JL (cid:15) (cid:15) (cid:31) (cid:127) (cid:47) (cid:47) C I × C JL (cid:15) (cid:15) E J (cid:47) (cid:47) C I × T J (cid:15) (cid:15) C I (cid:48) ⊕ C J (cid:48) E (cid:48) I (cid:48) (cid:47) (cid:47) T I (cid:48) × C J (cid:48) (cid:31) (cid:127) (cid:47) (cid:47) C I (cid:48) × C J (cid:48) E (cid:48) J (cid:48) (cid:47) (cid:47) C I (cid:48) × T J The morphism (cid:101) L descends to T I × C J because (cid:101) L ( (cid:101) σ ) ⊂ (cid:101) σ (cid:48) but since we donot make enough restriction on J and J (cid:48) , the morphism L has no reason todescend to a morphism C I × T J → C I (cid:48) × T J (cid:48) (like the simplicial case).Let T ∈ A and (cid:101) T cal m cal (cid:47) (cid:47) (cid:15) (cid:15) C I × T J T be an object of (cid:85) cal σ over T .Let (cid:98) T cal be the fibre product (cid:101) T m × (cid:101) E J ( C I ⊕ C J ) = (cid:110)(cid:0)(cid:101) t , z (cid:1) | m cal (cid:0)(cid:101) t (cid:1) = E J ( z ) (cid:111) .The group Z N − k × E I ( ker ( h σ )) acts on (cid:98) T cal by ( p , E I ( w , w )) · ( (cid:101) t , z , z ) = (( pr k ( p ) , E ( w , w )) · (cid:101) t , z + w , E ( w ) z ) where pr k : Z N − k = Z N − d ⊕ Z d − k → Z N − d is the projection (same defini-tion for pr k (cid:48) : Z N (cid:48) − k (cid:48) → Z N (cid:48) − d (cid:48) ).The linear map H χχ (cid:48) satisfies, for i ∈ {
1, . . . , k } , H χχ (cid:48) ( e i ) ∈ Z k (cid:48) ⊕ H χχ (cid:48) is of the form (cid:18) ∗ ∗ M (cid:19) ∈ M N , N (cid:48) ( Z ) (we do not careof the upper part since we apply the map E I ).In consequence, we can define the Z N (cid:48) − d (cid:48) × E (cid:48) ( ker ( h σ (cid:48) C )) -principal bun-dle (cid:98) T cal as (cid:101) T (cid:48) cal : = (cid:98) T cal × ( pr k (cid:48) ◦ M ) × E (cid:48) J ◦ L ( Z N (cid:48) − d (cid:48) × E (cid:48) ( ker ( h σ (cid:48) C ))) with associated equivariant map (cid:101) m cal : (cid:0)(cid:101) t , z , ( q , E (cid:48) ( w )) (cid:1) (cid:55)→ E (cid:48) J (cid:0) L ( z ) (cid:1) · E (cid:48) ( ϕ (cid:48) ( ⊕ q ) + w ) Hence, (cid:101) T (cid:48) cal (cid:15) (cid:15) m cal (cid:47) (cid:47) C I (cid:48) × T J (cid:48) T is an object of (cid:85) cal σ (cid:48) over T . Lemma 4.16.
The left square and the right square of the following diagram arecartesian : (cid:98) T cal (cid:47) (cid:47) (cid:15) (cid:15) C I × C JE J (cid:15) (cid:15) L (cid:47) (cid:47) C I (cid:48) × C J (cid:48) E (cid:48) J (cid:48) (cid:15) (cid:15) (cid:98) T cal × M × L ( Z N (cid:48) − d (cid:48) × E (cid:48) I (cid:48) ( ker ( h σ (cid:48) C ))) f (cid:15) (cid:15) g (cid:111) (cid:111) (cid:101) T cal m cal (cid:47) (cid:47) (cid:41) (cid:41) C I × T J (cid:47) (cid:47) C I (cid:48) × T J (cid:48) (cid:101) T (cid:48) cal (cid:101) m cal (cid:111) (cid:111) (cid:112) (cid:112) Twhere f ( (cid:101) t , z , q , E (cid:48) I (cid:48) ( t )) = ( (cid:101) t , z , pr k (cid:48) ( q ) , E (cid:48) ( t )) and g ( (cid:101) t , z , q , E (cid:48) I ( t )) = E (cid:48) I ( ϕ (cid:48) ( ⊕ q ) + t ) · L ( z ) Proof.
See proof of lemma 5.3 of [KLMV20] (cid:3)
We define the image of a morphism between objects of (cid:85) cal σ by the sameway as [KLMV20] in diagram 6.10.We get a stack morphism (cid:108) cal : (cid:85) cal σ → (cid:85) cal σ (cid:48) . Theorem 4.17. • The stack morphism (cid:108) cal is a toric morphism. • Let σ ⊂ R d , σ (cid:48) ⊂ R d (cid:48) , σ (cid:48)(cid:48) ⊂ R d (cid:48)(cid:48) be cones and ( L , H ) , ( L (cid:48) , H (cid:48) ) becalibrated quantum fans morphisms between the calibrated quantum fansassociated to it. Note (cid:108) , (cid:108) (cid:48) the toric morphisms associated to it and (cid:108) (cid:48)(cid:48) thetoric morphism associated to ( L (cid:48) ◦ L , H (cid:48) ◦ H ) . Then, (4.8) (cid:108) (cid:48)(cid:48) = (cid:108) (cid:48) ◦ (cid:108) • Let (cid:108) cal be a torus morphism between (cid:84) calh , I and (cid:84) calh (cid:48) , I (cid:48) and let ( L , H ) bethe induced linear morphisms. Then, (cid:108) cal extends as a toric morphism (cid:85) cal σ → (cid:85) cal σ (cid:48) if, and only if, ( L , H ) is a morphism of calibrated quantumfans.Proof. In [KLMV20], see the proof of lemma 5.4 for the second point andthe proof of theorem 5.5 and theorem 6.2 for the first and third points. (cid:3)
The two first points tell us that we defined a functor between the fullsubcategory of the quantum fans given by a cone and the category of affinequantum toric varieties. This functor is an equivalence of category thanksto the third one.Now, we can give the link between the construction of affine quantumvariety of this paper and the construction in [KLMV20] :
Proposition 4.18.
Suppose σ simplicial. We have a toric isomorphism : (cid:85) cal σ (cid:39) Q calk , d , P − χ ◦ ϕ (in the same manner as paragraph 6.3 in [KLMV20] ). ON-SIMPLICIAL QUANTUM TORIC VARIETIES 17
Proof. If σ is simplicial then h σ C is a monomorphism and h σ is an isomor-phism. The morphism ( h σ P χ , P χ ) is an isomorphism of calibrated quantumfans between the fan induced by the cone C k , d = Cone ( e , . . . , e k ) and thecalibration P − χ ◦ ϕ : Z N → P χ − ( h − σ ( Γ )) and the fan induced by σ and thecalibration h :Firstly, the following diagram commute : C N P − χ ϕ (cid:47) (cid:47) P χ (cid:15) (cid:15) C Ih σ C P χ (cid:15) (cid:15) C N h (cid:47) (cid:47) C d since ϕ = h − σ C ◦ h ◦ P χ . The map h σ C P χ sends C k , d onto σ since h σ C P χ ( e i ) = h σ C ( e χ ( i ) ) = v χ ( i ) ∈ σ The other points are obvious.Hence, it induces a toric isomorphism (thanks to theorem 4.17) (cid:85) cal σ (cid:39) Q calk , d , P − χ ◦ ϕ (cid:3) Quantum toric varieties.
Definition.
Let σ = σ I = Cone ( v i , i ∈ I ) and τ = σ J = Cone ( v j , j ∈ J ) be two cones of ∆ with a non-empty intersection. Note (cid:101) σ (resp. (cid:101) τ ) theassociated cone (see (4.1)) of R I (resp. R J ), h σ , h τ the associated grouphomomorphisms. In this subsection, we will see how to glue the quantumtoric varieties associated to them.We will suppose that σ and τ are dimension d but the following con-struction works for cones of any dimension.Let H στ : C I ∪ J → C d be the linear map such that H στ | C I = h σ C and H στ | C J = h τ C . The identity morphism (cid:84) cald , I → (cid:84) cald , I induces torus isomor-phisms(4.9) [ T I / Z N − d × E ( ker ( h σ C ))] (cid:39) [ T I ∪ J / Z N − d × E ( ker ( H στ C ))] (4.10) [ T I / Z N − d × ker ( h τ C )] (cid:39) [ T I ∪ J / Z N − d × E ( ker ( H στ C ))] Moreover, these morphisms extend to toric isomorphisms between (cid:85) cal σ =[ C I / Z N − d × E ( ker ( h σ C )] and [ C I × T J / Z N − d × E ( ker ( H στ ))] and between (cid:85) cal τ = [ C I / Z N − d × E ( ker ( h τ C ))] and [ T I × C J / Z N − d × E ( ker ( H στ ))] inthe same way as proposition 4.14.Note K I : = ( I ∪ J ) \ I and K J : = ( I ∪ J ) \ J .As the intersection σ ∩ τ is non-empty, the intersection { } × (cid:101) τ ∩ (cid:101) σ × { } in R I ∪ J is non-empty too. Hence, the classical theory gives us an opentoric subvariety (cid:83) στ of U { }× (cid:101) τ , (cid:83) τσ an open toric subvariety of U (cid:101) σ ×{ } and an toric isomorphism ϕ : (cid:83) στ → (cid:83) τσ . With some computations, we see that(with the decomposition C I ∪ J = C K J ⊕ C I ∩ J ⊕ C K I ) :(4.11) U (cid:101) σ ×{ } = C I × T K I , U { }× (cid:101) τ = T K J × C J ⊂ C I ∪ J and(4.12) ϕ = id : (cid:83) τσ = T K J × C I ∩ J × T K I → (cid:83) στ = T K J × C I ∩ J × T K I The identity descends to a stack isomorphism (thanks to the linear iso-morphism induced by the permutation χ (cid:48)− χ and the theorem 4.17) [ (cid:83) τσ / Z N − d × E ( ker ( H στ ))] (cid:39) [ (cid:83) στ / Z N − d × E ( ker ( H στ ))] (since the toric varieties (cid:83) στ , (cid:83) τσ are preserved by the actions).Now, thanks to the equality in (4.12) and the isomorphisms (4.9) and(4.10), we get a toric isomorphism (cid:103) I J between (cid:85) cal τσ : = [ T K J × C I ∩ J / Z N − d × E ( ker ( h σ C ))] (cid:44) → [ C I / Z N − d × E ( ker ( h σ C )] and (cid:85) cal στ : = [ C I ∩ J × T K I / Z N − d × E ( ker ( h τ C )] (cid:44) → [ C J / Z N − d × E ( ker ( h τ C )] . Remark . This transitions maps verify a cocycle condition since theidentity map does.With the previous discussion, we can define quantum toric varieties :
Definition 4.20.
Let T ∈ A . An object of (cid:88) cal ∆ , h cal , I over T is a covering ( T I : = T σ I ) of T indexed by the set of maximal cones I max together withan object (cid:101) T I m I (cid:47) (cid:47) (cid:15) (cid:15) C I × T K T I of [ C I × T K / Z N − d × E ( ker ( h σ I C ))]( T I ) for every σ I ∈ I max , satisfying forany couple ( I , I (cid:48) ) with non-empty intersection J (cid:103) II (cid:48) (cid:101) T I ⊃ J m I (cid:47) (cid:47) (cid:15) (cid:15) (cid:83) σ I σ I (cid:48) T I = (cid:101) T I (cid:48) ⊃ J m I (cid:48) (cid:47) (cid:47) (cid:15) (cid:15) (cid:83) σ I (cid:48) σ I T I (cid:48) where (cid:101) T I (cid:48) ⊃ J : = m − I ( (cid:83) σ I σ I (cid:48) ) and (cid:101) T I (cid:48) ⊃ J : = m − I (cid:48) ( (cid:83) σ I (cid:48) σ I ) A morphism of (cid:88) ∆ , h cal , I over a toric morphism T → S is defined as in[KLMV20] with the necessary modifications.4.2.2. Toric morphisms.
A morphism of quantum toric varieties (cid:88) cal ∆ , h cal , I → (cid:88) cal ∆ (cid:48) , ( h cal ) (cid:48) , I (cid:48) is a collection of compatible toric morphisms between theiraffine pieces. In other words, Definition 4.21.
A morphism (cid:88) cal ∆ , h cal , I → (cid:88) cal ∆ (cid:48) , ( h cal ) (cid:48) , I (cid:48) of calibrated quantumtoric varieties is a collection of toric morphisms (cid:108) σ : (cid:85) cal σ → (cid:85) cal σ (cid:48) for allmaximal cones of ∆ compatible with the glueing i.e. if the intersection ON-SIMPLICIAL QUANTUM TORIC VARIETIES 19 of σ and τ is not empty then the morphism (cid:108) σ (resp. (cid:108) τ ) restricts to amorphism (cid:85) cal τσ → (cid:85) cal τ (cid:48) σ (cid:48) (resp. (cid:85) cal στ → (cid:85) cal σ (cid:48) τ (cid:48) ) and such that the followingequality holds on (cid:85) cal τσ (4.13) (cid:103) (cid:48) σ (cid:48) τ (cid:48) (cid:108) σ = (cid:108) τ (cid:103) στ for each cones σ , τ with non-empty intersection.In paragraph 4.1.3, we proved a correspondence between the affinequantum toric varieties morphisms and the calibrated quantum morphisms.We will complete it :Let (cid:88) cal ∆ , h cal , I → (cid:88) cal ∆ (cid:48) , ( h cal ) (cid:48) , I (cid:48) be a morphism of calibrated quantum toricvarieties i.e. a collection of toric morphisms (cid:108) σ : (cid:85) cal σ → (cid:85) cal σ (cid:48) respecting theequalities (4.13) Let ( L σ , H σ ) be the linear maps associated to (cid:108) σ , ( g στ , k στ ) the linear isomorphisms associated to (cid:103) στ and ( g σ (cid:48) τ (cid:48) , k (cid:48) σ (cid:48) τ (cid:48) ) the linear iso-morphisms associated to (cid:103) (cid:48) σ (cid:48) τ (cid:48) . Then, thanks to the second point of theo-rem 4.17, the equations (4.13) becomes :(4.14) g (cid:48) σ (cid:48) τ (cid:48) ◦ H σ = H τ ◦ g στ and k (cid:48) σ (cid:48) τ (cid:48) ◦ L σ = L τ ◦ k στ Hence, we can glue these calibrated quantum fans morphisms into onebetween ( ∆ , h cal , I ) → ( ∆ (cid:48) , (cid:0) h cal (cid:1) (cid:48) , I (cid:48) ) .Conversely, a calibrated quantum fans morphism ( ∆ , h cal , I ) → ( ∆ (cid:48) , (cid:0) h cal (cid:1) (cid:48) , I (cid:48) ) defines toric morphism between affine calibrated quantum toric varietiesverifying (4.13) i.e. defines a toric morphism (cid:88) cal ∆ , h cal , I → (cid:88) cal ∆ (cid:48) , ( h cal ) (cid:48) , I (cid:48) We proved our main theorem :
Theorem 4.22.
The correspondence ( ∆ , h cal , I ) → (cid:88) cal ∆ , h cal , I is functorial anddefines an equivalence of categories between the category of calibrated quantumfan and the category of calibrated quantum toric varieties.
5. GIT- like construction
In this section, we will discuss the realization of a quantum toric varietyas a global quotient stack.We can build the glueing of quantum toric varieties in an another waythan the previous section :Let (cid:83) be the glueing of the toric varieties C I and C J along the intersection { } × (cid:101) τ ∩ (cid:101) σ × { } . Then, we define the glueing of (cid:85) cal σ and (cid:85) cal τ is thequotient stack [ (cid:83) / Z N − d × E ( ker ( H στ ))] . Then, we deduce Theorem 5.1.
Let A be the set A : = (cid:83) σ I ∈ ∆ I. Let (cid:101)
A a subset of {
1, . . . , N } \ I such that ( v i , i ∈ (cid:101) A ) is free and such that C d = Vect ( v i , i ∈ A ) ⊕ Vect ( v i , i ∈ (cid:101) A ) Let h A : Z A ∪ (cid:101) A → Γ be the group homomorphism defined by h ( e i ) = v i fori ∈ A ∪ (cid:101) A. Note (cid:83) A the toric variety given by the associated fan (see (4.1) ) of ∆ in R A given by ∆ h ∩ R A = { τ | ∃ σ ∈ ∆ , τ (cid:22) (cid:98) σ ∩ R A } . We define an action of Z N − d × E ( ker ( h A )) on (cid:83) A × T (cid:101) A in the same way as4.1.2. Then, (cid:88) cal ∆ , h cal , I (cid:39) [ (cid:83) A × T (cid:101) A / Z N − d × E ( ker ( h A ))] as stacks Moreover, in the same way as in the beginning of subsection 4.2, wehave a toric isomorphism [ C I × T J / Z N − d × ker ( h σ C )] (cid:39) [ C I × T I c / Z N − d × ker ( h cal C )] Hence, we get a GIT-like realization of (cid:88) cal ∆ , h cal , I : Theorem 5.2.
Let (cid:83) be the toric variety associated to the associated fan ∆ h . Then,we have a stack isomorphism (5.1) (cid:88) cal ∆ , h cal , I (cid:39) [ (cid:83) / Z N − d × E ( ker ( h cal C ))] which restricts to a torus isomorphism between the associated quantum torus oneach affine chart. By contrast to the simplicial case, we cannot realize (cid:88) cal ∆ , h , I as a quotientof a toric variety by an action of C N − d via Gale transform (the "quantumGIT" of [KLMV20]).Indeed, let k : C N − d → C N be the map induced by a Gale transform(see [GKKZ03] for the details) of the family ( h cal ( e i ) , i ∈ [[ N ]]) i.e. a mapsuch that the sequence0 (cid:47) (cid:47) C N − d k (cid:47) (cid:47) C N h cal C (cid:47) (cid:47) C d (cid:47) (cid:47) (cid:88) cal ∆ , h cal , I and [ (cid:83) / C N − d ] are isomorphic (where C N − d acts on (cid:83) through E ◦ k ) in order to generalize the theorem 7.6 of[KLMV20]).For each σ ∈ ∆ , the corresponding open substack of (cid:88) cal ∆ , h cal , I is (cid:85) cal σ =[ C I × T J / Z N − d × E ( ker ( h σ C ))] and the substack of [ (cid:83) / C N − d ] is (cid:85) (cid:48) σ : =[ C I × T I c / C N − d ] where I c = {
1, . . . , N } \ I . Proposition 5.3.
The stacks (cid:85) cal σ and (cid:85) (cid:48) σ are not isomorphic if σ is not simplicial.Proof. The groupoid associated to (cid:85) cal σ and the groupoid associated to (cid:85) (cid:48) σ cannot be Morita-equivalent since their isotropy groups (i.e. the stabilizeron a point by the action) are not isomorphic (see theorem 4.4 of [Xu04]).More precisely, the stabilizer of the action of Z N − d × E ( ker ( h σ C )) at eachpoint of ( C (cid:101) I ⊕ ) × T J is E ( ker ( h σ )) × ker ( h cal ) and there is no point of C I × T I c with an isomorphic stabilizer group for the C N − d -action. Indeed,the different stabilizer groups for the C N − d -action are k − ( ker ( h σ C ) ∩ C K + ker ( h cal )) at C (cid:101) I ⊕ ( ker ( h σ ) ∩ { } K ) × T I c for ∅ (cid:54) = K (cid:32) I and k − ( ker ( h σ C ) + ker ( h cal )) at ( C (cid:101) I ⊕ { } ) × T I c . These groups cannot be isomorphic to E ( ker ( h σ )) × ker ( h cal ) (think of the lack of isomorphism between C n and ( C ∗ ) n × Z n ). (cid:3) ON-SIMPLICIAL QUANTUM TORIC VARIETIES 21
Conversely, if we replace the construction of quantum toric varietiesby the quotient stack of (cid:83) by the C N − d -action, the construction wouldnot be fonctorial. Indeed, its "torus" is the quotient stack [ T N / C N − d ] which is not isomorphic to the calibrated quantum torus (cid:84) calh , I (for thesame reason as the proof of 5.3) and hence, this torus is not entirelycharacterized by the calibration (up to isomorphism). Therefore, we can-not associate a stack morphism to a calibrated quantum fan morphism ( L : R d → R d (cid:48) , H : R N → R N (cid:48) ) .6. F orgetting calibration and gerbe structure We can associate to each affine quantum toric variety (cid:85) cal σ = [ C I × T J / Z N − d × E ( ker ( h σ C ))] a "non-calibrated quantum toric variety" (cid:85) σ : =[ C I × T J / E ( h − σ C ( Γ ))] .More precisely, note Ξ be the kernel of h cal and let T be an object of A and ( (cid:101) T cal , m cal ) be an object of (cid:85) cal σ ( T ) .Since h × id : Z N − d × E ( ker ( h σ C )) → Z N − d × E ( ker ( h σ C )) Ξ × = E ( Γ ) × E ( ker ( h σ C )) = E ( h − σ C ( Γ )) is a Ξ -covering then (cid:101) T = (cid:101) T cal / Ξ → T is a E ( h − σ C ( Γ )) -principal bundle.Note m : [ (cid:101) t cal ] ∈ (cid:101) T (cid:55)→ m cal ( (cid:101) t cal ) ∈ C I × T J . This map is well-defined : m cal (( ξ , 0 ) · (cid:101) t cal ) = E ( ϕ ( ξ , 0 )) m cal ( t cal ) = m cal ( t cal ) We can define morphisms of non-calibrated affine quantum toric varietiesin the same way as 4.15 :
Definition 6.1.
A toric morphism between two non-calibrated affine quan-tum toric varieties (cid:85) σ = [ C I × T J / G ] and (cid:85) σ = [ C I (cid:48) × T J (cid:48) / G ] is a stackmorphism [ C I × T J / G ] → [ C I (cid:48) × T J (cid:48) / G ] which restricts to a presentednon-calibrated torus morphism [ T I × T J / G ] → [ T I (cid:48) × T J (cid:48) / G ] We can see that morphism of affine quantum toric varieties descends toquotient and induce a morphism of non-calibrated affine quantum toricvarieties. Hence, we have defined a functor f from the category of affinequantum toric varieties to the category of non-calibrated affine quantumtoric varieties.We can follow the proof of 4.17 in order to see that a morphism ( L , H ) of calibrated quantum fan morphism (we need the two morphisms due tothe presence of the kernel of h σ C ) leads to a morphism (cid:108) of non-calibratedaffine quantum toric varieties. Lemma 6.2.
This functor coincide with the functor f of [KLMV20] (section 6.2)on simplicial quantum toric variety. Moreover, the functor f induces (with lemma 3.5) a functor (cid:101) f on thecategory of presented quantum tori defined by (cid:101) f ( (cid:84) calh , I , h (cid:48) : Z N → G , I (cid:48) , L , H , s ) = ( (cid:84) d , Γ , L ) and (cid:101) f ( L , H , S , L (cid:48) , H (cid:48) , S (cid:48) ) = ( L , L (cid:48) ) Lemma 6.3.
We have a commutative diagram (cid:85) cal σ (cid:108) cal (cid:47) (cid:47) f (cid:15) (cid:15) (cid:85) cal σ (cid:48) f (cid:15) (cid:15) (cid:85) σ (cid:108) (cid:47) (cid:47) (cid:85) σ (cid:48) We can adapt the definition of quantum toric varieties to the non-calibratedcase :
Definition 6.4.
Let T ∈ A . An object of (cid:88) ∆ , Γ over T is a covering ( T I : = T σ I ) of T indexed by the set of maximal cones I max together with the image byf of an object (cid:101) T I m I (cid:47) (cid:47) (cid:15) (cid:15) C I × T K T I of [ C I × T K / Z N − d × E ( ker ( h σ I C )]( T I ) for every σ I ∈ I max , satisfying forany couple ( I , I (cid:48) ) with non-empty intersection J f (cid:103) II (cid:48) (cid:101) T I ⊃ J m I (cid:47) (cid:47) (cid:15) (cid:15) (cid:83) σ I σ I (cid:48) T I = f (cid:101) T I (cid:48) ⊃ J m I (cid:48) (cid:47) (cid:47) (cid:15) (cid:15) (cid:83) σ I (cid:48) σ I T I (cid:48) where (cid:101) T I (cid:48) ⊃ J : = m − I ( (cid:83) σ I σ I (cid:48) ) and (cid:101) T I (cid:48) ⊃ J : = m − I (cid:48) ( (cid:83) σ I (cid:48) σ I ) A morphism of (cid:88) ∆ , Γ over a toric morphism T → S is defined by apply-ing f to a morphism of (cid:88) cal ∆ , h cal , I over this morphism. Theorem 6.5.
The quantum toric variety (cid:88) cal ∆ , h , I is a gerbe over (cid:88) ∆ , Γ with band Z rk ( Ξ ) . In particular, if Ξ = then these two stacks are isomorphic. This structure of gerbe induces a T rk ( Ξ ) -bundle over (cid:88) Γ , ∆ up to homo-topy. The end of this section will be devoted to describe it :Let σ = σ I ∈ ∆ be a maximal cone, h σ C be the C -linear morphismassociated to it, ψ : C I / ker ( h σ C ) → C d induced by h σ C .Let (cid:101) I be a subset of I such that ( h ( e i ) , i ∈ (cid:101) I ) is a basis of Vect ( σ ) , J be a setof cardinal d − dim ( σ ) such thatVect ( h cal ( e i ) , i ∈ I ∪ J } = C d ,let χ ∈ S N be a permutation such that(6.1) χ ( {
1, . . . , dim ( σ ) } ) = (cid:101) I and χ ( { dim ( σ ) +
1, . . . , d } ) = J and P χ ∈ GL N ( R ) be the map associated to χ . ON-SIMPLICIAL QUANTUM TORIC VARIETIES 23
Let ϕ = ( ϕ , ϕ ) be the linear map ψ − hP χ : C N → C I / ker ( h σ ) × C J .Thanks to the conditions (6.1), we get the following diagram C N ϕ (cid:47) (cid:47) ( ψ − , id C N − d ) ◦ E k (cid:15) (cid:15) C I × C J / ker ( h σ ) E I (cid:15) (cid:15) T I / E ( ker ( h σ C ) × C J × C N − d (cid:127) (cid:95) (cid:15) (cid:15) ϕ (cid:47) (cid:47) T I × C J / E ( ker ( h σ C )) (cid:127) (cid:95) (cid:15) (cid:15) (cid:2) C I / E ( ker ( h σ C ) (cid:3) × C J × C N − d ϕ (cid:47) (cid:47) ( id , E , id ) (cid:15) (cid:15) (cid:2) C I × C J / E ( ker ( h σ )) (cid:3) E J (cid:15) (cid:15) (cid:2) C I / E ( ker ( h σ C ) (cid:3) × T J × C N − dE d (cid:15) (cid:15) (cid:98) ϕ (cid:47) (cid:47) (cid:2) C I × T J / E ( ker ( h σ )) (cid:3) (cid:15) (cid:15) (cid:2) C I / E ( ker ( h σ C ) (cid:3) × T J × T N − d ϕ cal (cid:47) (cid:47) (cid:85) cal σ where for all ( z , z ) ∈ T I × C J , w ∈ C N − d ϕ ([ z ] , z , w ) = ([ E ( ϕ ( ⊕ w )) z ] , ϕ ( ⊕ w ) + z ) and z ∈ C I × T J , w ∈ C N − d , (cid:98) ϕ ([ z ] , z , w ) = ([ E ( ϕ ( ⊕ w )) z ] (we can translate this equality in terms of principal bundles)The group Z N − d acts on (cid:2) C I / E ( ker ( h σ C )) (cid:3) × T J × C N − d by translationon the last factor and on (cid:2) C I / E ( ker ( h σ C )) (cid:3) × T J × C N − d through the mor-phism ϕ . The morphism (cid:98) ϕ is equivariant for these two actions. Hence, (cid:98) ϕ descends to a stack morphism (cid:2) C I / E ( ker ( h σ C ) (cid:3) × T J × T N − d → (cid:85) cal σ In the same manner, the projection on the first factor (cid:16) [ C I / E ( ker ( h σ C ))] × T J (cid:17) × C N − d → [ C I / E ( ker ( h σ C ))] × T J is equivariant for the action (on the source) of Z N − d defined by, for p ∈ Z N − d , ( z , w ) ∈ C I × T J (6.2) p · ( z , w ) = ( E ( ϕ ( ⊕ p )) z , w + p ) (it is well-defined since the actions are multiplicative) and the action of Z N − d on the target defined by, for p ∈ Z N − d and for ( z , w ) ∈ ( C I × T J ) (6.3) p · z = E ( ϕ ( ⊕ p )) z Hence, the projection descends to a stack morphism :(6.4) (cid:0) [ C I / E ( ker ( h σ C ))] × T J (cid:1) × C N − d π (cid:47) (cid:47) (cid:15) (cid:15) [ C I / E ( ker ( h σ C ))] × T J (cid:15) (cid:15) (cid:2) ( C I × T J ) × C N − d / E ( ker ( h σ )) × Z N − d (cid:3) (cid:112) (cid:47) (cid:47) (cid:85) cal σ Moreover, we have the following commutative diagram ( (cid:2) C I / E ( ker ( h σ C )) (cid:3) × T J ) × C N − d ( (cid:98) ϕ , id ) − (cid:15) (cid:15) π (cid:47) (cid:47) (cid:2) C I × T J / E ( ker ( h σ )) (cid:3) [ C I / E ( ker ( h σ C ))] × T J × C N − d (cid:98) ϕ (cid:51) (cid:51) Since the morphism ( (cid:98) ϕ , id ) − is equivariant for the action (6.2) and for thetranslation in the last factor. Hence, we get the following commutativediagram(6.5) (cid:2) ( C I × T J ) × C N − d / E ( ker ( h σ )) × Z N − d (cid:3) ( (cid:98) ϕ , id ) − (cid:15) (cid:15) (cid:112) (cid:47) (cid:47) (cid:85) cal σ [ C I / E ( ker ( h σ C ))] × T J × T N − d ϕ cal (cid:52) (cid:52) The diagrams (6.4) and (6.5) can be seen as trivialization of the mor-phism ϕ cal . Thus, we can see ϕ cal as a C N − d -fibre bundle.Note (cid:88) ∆ the stack over A given by the descent data of the family ofstacks [ C I / E ( ker ( h σ C ))] × T J × T N − d indexed by every maximal cones of ∆ (in the same manner as 4.2). By functoriality, we get Theorem 6.6.
The map ϕ cal : (cid:88) ∆ → (cid:88) cal ∆ , h , I is a C N − d -principal bundle (in thesense that this morphism can be locally trivialized on the affine pieces, the trivial-ization map induced the identity on the factors C N − d and the transition maps aregiven by the action of C N − d ) This statement gives further details on the theorem 6.21 of [KLMV20].
Proof.
It remains us to explain to the action induced by the transition maps.By the definitions of (cid:88) ∆ and (cid:88) cal ∆ , h , I , we can restrict us to the simplicial fansi.e. (cid:88) ∆ = (cid:83) .Each cone σ I of ∆ induces an action of the group C N − d on each variety C I × T J × C ( I ∪ J ) c ⊂ (cid:83) . Namely, it is the induced C N − d -action by the actionby translation on the last factor of C I × T J × C N − d and the isomorphism ( (cid:98) ϕ I , id ) ◦ ( id , P χ I | C N − d ) − i.e.(6.6) λ · ( z , z , w ) = ( E ( − (cid:101) ϕ I ( λ )) z , E ( − (cid:101) ϕ I ( λ )) z , w + P χ I λ ) where (cid:101) ϕ I = ( (cid:101) ϕ I , (cid:101) ϕ I ) : C N − d → C I × C J is the restriction of ϕ I on C N − d .Thus, the morphism (cid:99) ϕ I ◦ ( id , P χ I | C N − d ) − : C I × T J × C ( I ∪ J ) c → C I × T J isa C N − d trivializable principal bundle. Hence, it has a global cross section s σ : C I × T J → C I × T J × C ( I ∪ J ) c which is defined by s σ ( x , y ) = ( x , y , 0 ) This morphism descends as a morphism (cid:85) cal σ → C I × T J × T N − d .In order to conclude, we have to find, for each cones σ = σ I , τ = σ I (cid:48) with a non-empty intersection, a morphism t στ : (cid:85) cal στ → C n − d such that(6.7) s σ = t στ · s τ ON-SIMPLICIAL QUANTUM TORIC VARIETIES 25
Note ϕ I = ψ − I hP χ I and ϕ I (cid:48) = ψ − I (cid:48) hP χ I (cid:48) the calibration associated, re-spectively, to σ and τ . Recall the commutative diagram used in the defini-tion of quantum toric varieties : C I ∪ J ψ I (cid:47) (cid:47) C d ψ − I (cid:48) (cid:47) (cid:47) C I (cid:48) ∪ J (cid:48) ψ − I ( Γ ) ψ I (cid:47) (cid:47) (cid:63)(cid:31) (cid:79) (cid:79) Γ ψ − I (cid:48) (cid:47) (cid:47) (cid:63)(cid:31) (cid:79) (cid:79) ψ − I (cid:48) ( Γ ) (cid:63)(cid:31) (cid:79) (cid:79) Z n ϕ I (cid:79) (cid:79) P χ I (cid:47) (cid:47) Z nh (cid:79) (cid:79) P − χ I (cid:48) (cid:47) (cid:47) Z n ϕ I (cid:48) (cid:79) (cid:79) Since ψ − I (cid:48) ψ I ϕ I = ϕ I (cid:48) ( P − χ (cid:48) I P χ I ) then for all point m of Z d ,(6.8) ψ − I (cid:48) ψ I ( P χ m ) = ϕ I (cid:48) ( P − χ I (cid:48) P χ I )( m ) = ( Id + (cid:101) ϕ I (cid:48) ) ◦ ( P − χ I (cid:48) P χ I )( m ) Since we consider the simplicial case, we can use the quasifold formalism(like the equation (6.33) of [KLMV20]) :The transition map between (cid:85) cal τσ and (cid:85) cal στ is [ z P χ ] ∈ (cid:85) cal τσ (cid:55)→ (cid:104) z ψ − I (cid:48) ψ I P χ (cid:105) ∈ (cid:85) cal στ Note K II (cid:48) the set P − χ I (cid:48) ( I ) \ {
1, . . . , d } . Then, the maps t στ : z (cid:55)→ (cid:2) z K II (cid:48) (cid:3) verify the equality (6.7) thanks to the equalities (6.6) and (6.8). (cid:3) As (cid:88) cal ∆ , h , I is a gerbe over (cid:88) ∆ , Γ with band Z rk ( Ξ ) then Theorem 6.7.
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