NON-SIMPLICIAL QUANTUM TORIC VARIETIES

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NON-SIMPLICIAL QUANTUM TORIC VARIETIES

Antoine Boivin

To cite this version:

Antoine Boivin. NON-SIMPLICIAL QUANTUM TORIC VARIETIES. 2020. �hal-02888226�

ANTOINE BOIVIN

Abstract. In this paper, we define quantum toric varieties associated to an arbitrary fan in a finitely generated subgroup of someR^d gener- alizing the article [KLMV20] of Katzarkov, Lupercio, Meersseman and Verjovsky.

Contents

1. Introduction 1

2. Notations and conventions 3

2.1. Stacks 3

2.2. Fans 3

3. Quantum tori 4

4. Definition of (non-simplicial) quantum toric varieties 7

4.1. Affine quantum toric varieties 7

4.2. Quantum toric varieties 17

5. GIT-like construction 19

6. Forgetting calibration and gerbe structure 21

References 25

1. Introduction

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. Such a variety can be described by a fan of rational strongly convex polyhedral cones on a latticeΓ(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 us a dictionary between geometric properties of toric varieties and combina- torial properties of fans making of toric varieties one of the most studied class of complex algebraic varieties.

However, in the classical theory, this fan has to be rational. Therefore the toric varieties are rigid i.e. we can not deform them. Indeed, if we deform a lattice, it can become not discrete (for instance, the group Z+_αZ is discrete if αis rational but is dense inRifαis 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 2, 2020.

1

each cones of the fan are R-linearly independent) on a finitely generated (possibly irrational) subgroup of some R^d (named "quantum fan"). When the fan is rational, one recovers the classical toric variety but the case of irrational simplicial fans is also covered. This construction is functorial and defines an equivalence of categories between the category of quantum toric varieties and the category of quantum simplicial fans (see theorems 5.18 and 6.24 of [KLMV20]).

Now, the simplicial fans form only a small part of all the fans. In the classical theory, they correspond to the toric varieties which are orbifold (i.e. with cyclic singularities). Hence, the restriction to simplicial fans is strong.

In this paper, we extend the construction of [KLMV20] to the general case i.e. we define the quantum toric variety associated to an arbitrary fan.

The non-simplicial case brings new problems. Indeed, in the simplicial case, 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 is a standard cone Cone(e₁, . . . ,e_k)⊂_R^d(where{e₁, . . . ,e_d}is the canonical basis ofR^d). Moreover, we can easily find the faces of these cones: it is the cones generated by a subfamily of{e₁, . . . ,e_k}. This basic fact is repeatedly used in construction of [KLMV20]. It is not the case for non-simplicial case anymore. 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 generated by a subfamily of the 1-cones is not necessarily a face of the cone since the cone can have an arbitrarily big number of 1-cones.

One of the key technical points is to replace the calibration h of the group Γused in [KLMV20] by a calibration ϕof a group isomorphic toΓ but which is a subgroup of an higher dimensional R^p for each maximal cone of the fan (pis the number of 1-cones of the maximal cone).

In section 3, we give a suitable definition of quantum tori in order to deal with the non-simplicial case and more precisely, these quantum tori encode the change of calibration of the last paragraph.

In section 4, we define the affine quantum toric variety (which will be calibrated 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 toric variety with the descent data of the affine pieces associated to a quantum fan. In particular, this construction works for simplicial fans and coincide with 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 in section 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

epimorphism ϕby the action of ϕ(_Z^N−d)and hence removing the ineffec- tivity of the action) in section 6.

2. Notations and conventions

2.1. Stacks. We will take the same conventions on stacks as [KLMV20] : LetAbe the category of affine toric varieties with toric morphisms. We endow this category a structure of site with the coverings{U_i ,→ T}_{i∈{1,...,n}}, whereU_i are open subsets ofT andT=^Sn_i=1U_i.

Let G be the category of complex analytic spaces with an action of an abelian complex Lie groupGwith an open orbit isomorphic toG.

Definition 2.1. LetHbe an abelian Lie group andXbe an object ofGwith an action of Hcommute with the action ofG.

The stack [X/H]is the stack overAwhose objects overT are H-principal bundle Te → T (inG) with an H-equivariant morphism Te → Xand mor- phisms over S → T are a bundle morphism Se → Te compatible with the equivariant maps.

All the stacks in this paper will be of this form or given by the descent data 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 that Vect_R(_Γ) =_R^d. A calibration ofΓis given by :

• A group epimorphism h: Z^N →_Γ

• A subsetI ⊂ {1, . . . ,N}such that Vect_C(h(e_j),j∈ I) =/ _C^d (this is the set of virtual generators)

This is a standard calibration if Z^d ⊂ _Γ,h(e_i) =e_i fori=_{1, . . . ,}dandI _is of the form {n− |I |+1, . . . ,n}

Definition 2.3. A calibrated quantum fan (_∆,h,I)_inΓis the data of

• a collection ∆ of strongly convex polyhedral cones generated by elements ofΓsuch that every intersection of cones of∆is a cone of

∆, every face of a cone of∆is a cone and {0}is a cone of∆.

• a standard calibrationh withI its set of virtual generators

• A set of generators A i.e. a subset of{1, . . . ,N} \ I such that the 1-cone generated by theh(e_i)fori∈ Aare 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 inΓ. Note∆h the fan inZ^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) inΓand(_∆⁰,h⁰,I⁰)is a pair of linear morphisms(L:R^d →_R^d0,H:R^N → R^N0)where

• The diagram

(2.2) R^{n hR //}

H

R^d

L

R^{n0 h0R //} R^d0

commutes (where h_R (resp. h⁰_R) is the R-linear map associated toh (resp. toh⁰))

• L(_Γ)⊂ _Γ⁰

• If σ ∈ _∆ then it exists σ⁰ ∈ _∆⁰ such that L(σ) ⊂ σ⁰ (thanks to the first point, the same statement is true for H)

• If H(e_i) ∈ Cone(e_j,j ∈ I) ∈ _∆h then H(e_i) ∈ ^L_j∈I0Nej (same statement is true for L and the vectors h(e_i), thanks to the first point)

• For alli∈ I/ ,H(e_i)∈^L_j/∈I0Ze_j

• There exists a maps:I → I⁰ such that for alli∈ I,H(e_i) =e_s(i) Remark2.5. We can define non-calibrated counterpart of these definitions by replacing the data of the calibration by the data of the 1-cones h(e_i)_, i∈ {1, . . . ,N}

3. Quantum tori

Let Γa finitely generated subgroup ofR^d andh :Z^N →_Γa calibration of Γ(noteI the set of virtual generators).

The standard calibrated quantum torus associated to these data is the quo- tient stackT_h,^cal_I = [_C^d/Z^N]whereZ^N acts onC^dthrough the morphismh.

However, since a non-simplicial cone can have more thandgenerators, we have to consider quantum tori as a quotient stack of a higher-dimensional space C^p. In order to do this, we will replace the group Γand 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 of the group Γ) with the data of a stack isomorphism, respecting the virtual generators, with such quotient stack. More precisely,

Definition 3.1. A presented calibrated quantum torus is a 6-uple(T_h,^cal_I,ϕ: Z^N → G ⊂ _C^p,I⁰,L,H,s) _where ϕ is a calibration of the group G (with I⁰ 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 → I⁰ is a bijection such that :

• L(G) =_ΓandL_|G :G→_Γis a group isomorphism.

• The diagram Z^{N H //}

ϕ

Z^N

h

G ^{L|G //}Γ

is commutative.

• For alli∈ I,H(e_i) =e_s(i) and for alli∈ I/ ,H(e_i)∈^L_j∈I/ 0Zej

The morphism ϕ is the calibration of this presented calibrated quantum torus.

With these data, we can define the quotient stack [_C^p_/Z^N×ker(L⊗_R id_C)]_{, whereZ}^N×ker(L⊗_Rid_C)_{acts onC}^p _through

(m,w)·z= z+ϕ(m) +w, Moreover, these data encode a stack isomorphism

[_C^p_/Z^N×ker(L⊗_Rid_C)]' T_h,^cal_I.

If L is a linear isomorphism then this stack isomorphism is a calibrated torus 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 (T_d,Γ,L)whereLis a linear epimorphismR^p →_R^d

In particular, the linear morphism descends to a stack isomorphism [_C^p/L⁻_C¹(_Γ)]' T_d,Γ

(where L_C = L⊗_Rid_C and L⁻_C¹(_Γ) acts onC^p by translations) which is a quantum torus isomorphism (in the sense of [KLMV20]) if L is a linear isomorphism.

In both cases, we can define a multiplicative form of the torus thanks to the exponential map

E:(z₁, . . . ,z_p)∈_C^p7→ (e^2iπz1, . . . , e^2iπzp)∈_T^p := (_C^∗)^p

(1) In the non-calibrated case, we have an isomorphism (see diagram 3.12 of [KLMV20]) :

[_C^p/L⁻_C¹(_Γ)]'[_T^p/E(L⁻_C¹(_Γ))]

where E(L⁻_C¹(_Γ))acts multiplicatively onT^p.

(2) In the calibrated case, we have to suppose hstandard i.e. we sup- pose that there is a subset eI = {i₁, . . . ,i_d} ⊂ {1, . . . ,p} such that Z^eI ⊂ G and h(e_k) = e_ik for k = 1..d. Then, the exponential map gives us :

[_C^p_/Z^N×ker(L_C)]'[_T^p_/Z^N−d×E(ker(L_C))]

whereZ^N−d×E(ker(L_C))acts onT^p through (m,E(w))·z= E(ϕ(0⊕m) +w)z

Definition 3.3. A morphism of presented non-calibrated quantum tori (T_d,Γ,L)→(T_d⁰_,Γ⁰,L⁰)

or presented torus morphism is a couple(L,L⁰)of linear morphisms such that

C^d

L

C^p/ ker(L_C)

[L_C]

oo

L⁰

C^d0 C^p0/ ker(L⁰_C)

[L⁰_C]

oo

where [L]is the morphism induced by Lon the quotient.

Definition 3.4. A morphism of presented calibrated quantum tori

(T_h,^cal_I,ϕ:Z^N → G⊂_C^p,Ie,L,H,s)→(T_h^cal0,I⁰,ϕ⁰ :Z^N0→ G⁰ ⊂_C^p0,Ie⁰,L⁰,ϕ,s⁰) or presented calibrated torus morphism is a 6-uple(L,H,S,L⁰,H⁰,S⁰)of morphisms such that

• (L,H,S) induces a torus morphism T_h,^cal_J → T_h^cal0,J⁰ as defined in [KLMV20] (definition 3.11),

• for all j∈ Ie,H⁰(e_j) =e_S⁰_(j) and for allj∈/Ie, H⁰(e_j)∈ ^M

k/∈Ie⁰

Zek

• The following diagrams commute : C^N

H⁰

C^N

oo H

H

h //C^d

L

C^p/ ker(L_C)

[L_C]

oo

L⁰

C^N0 C^N0

H⁰

oo

h⁰

//C^d0 C^p0/ ker(L⁰_C)

[L⁰_C]

oo

I ^{s //}

S

Ie

S⁰

I^{0 s0 //}Ie⁰

In particular, we have the following commutative diagram showing the relation between the different calibrations :

Z^N

ϕ

H //Z^{N H //}

h

Z^N0

h⁰

H⁰⁻¹ //Z^N0

ϕ⁰

G L // Γ

L //Γ⁰

(L⁰_|G0)⁻¹

//G⁰

We can reformulate this in term of quotient stack : Lemma 3.5. We use the same notations as definition 3.4.

Let l⁰ : [_C^p_/Z^N×ker(L_C)] → [_C^p0_/Z^N0×ker(L⁰_C)] be the stack morphism described by the linear morphisms (L⁰,H⁰)and l^cal : T_d,^cal_Γ → T_d^cal0,Γ⁰ be the stack morphisms described by the linear morphisms(L,H). Then, the diagram

[_C^p_/Z^N×ker(L_C)] ^{l0 //}

'

[_C^p0_/Z^N0×ker(L⁰_C)]

'

T_d,Γ^{cal lcal //}T_d^cal0,Γ⁰

commutes (where the isomorphisms are the stack isomorphisms encoded by the presentation of the presented calibrated quantum tori).

The presented quantum tori are (essentially) not new quantum tori. In fact, all presented quantum tori with the same underlying quantum tori are isomorphic :

Lemma 3.6. (T_h,^cal_I,ϕ : Z^N → G ⊂ _C^p,I⁰,L,H,s) and (T_h,^cal_I,h : Z^N → Γ,I,id,id,id)(resp. (T_d,Γ,L)and(T_d,Γ,id)) are isomorphic.

Proof. An isomorphism is given by(id,id,id,[L],H,s)(resp. (id,[L])).

Proposition 3.7. The forgetful functor

(T_h,^cal_I,ϕ:Z^N → G,I⁰,L,H,s)→ T_h,^cal_I,

resp. (T_d,Γ,L)→ T_d,Γ, is an equivalence of categories between the category of pre- sented calibrated quantum tori and the category of standard calibrated quantum tori, resp. between the category of non-calibrated quantum tori and the category of non-calibrated standard quantum tori.

Proof. Use lemma 3.6

4. Definition of(non-simplicial)quantum toric varieties Let Γ be a finitely generated subgroup ofR^d such that Vect_R(_Γ) =_R^d, (_∆,h^cal: Z^N →_Γ,I)a standard calibrated quantum fan (withI its set of virtual generators). In order to define a quantum variety associated to this fan, we will describe the affine quantum variety associated to each cone of this fan. The crucial point is to replace the calibrationhofΓby an adapted calibration ϕfor each maximal coneσof the fan∆.

After that, we will define the quantum toric varieties with the descent data of affine pieces as [KLMV20].

At the end of this section, we will prove that the correspondence (_∆,h^cal:Z^N →_Γ,I)7→X_∆,h^cal_cal,I

is an equivalence of categories.

4.1. Affine quantum toric varieties. Let

σ =σ_I =Cone(v_i1 := h^cal(e_i1), . . . ,v_ip := h^cal(e_ip))

(with I :={i₁, . . . ,i_p} ⊂ {_{1, . . . ,}N})be a strongly convex cone of∆. There are two possibilities : σ is of maximal dimension i.e. of dimension d or not. In the latter, we have to do a choice of a completion of Vect_R(σ) in

R^d. Fortunately, all obtained quantum varieties are isomorphic and this isomorphism respects a cocycle condition.

4.1.1. Cones of dimension d. Supposeσis a cone of dimensiondi.e. Vect_R(σ) = R^d.

Note h_σ :Z^I → _Γthe restriction ofh^cal onZ^I,h_σC :C^I → _C^d theC-linear map associated to it andbσ the cone ofR^N defined by

(4.1) bσ=Cone(e_i,i∈ I) andeσits restriction onR^I

Let B = (v_i,i ∈ eI) a subfamily of (v₁, . . . ,v_p) which is a basis of C^d. Then, we will consider the decomposition C^I = _C^eI⊕ker(h_σC). The map h_σC induces a linear isomorphismψbetweenC^eI andC^d.

Let χ∈ S_N be a permutation such thatχ({1, . . . ,d}) =eI.

Definition 4.1. The linear morphism ϕ:C^N →_C^eI ,→_C^I defined by e_k 7→ψ⁻¹(h_C^cal(e_χ(k)))

is called calibration associated to σ(andB andχ)

We can think this morphism as a calibration induced by the calibra- tion h on C^I. Indeed, the image of Z^d⊕0 by the morphism ϕ is Z^eI (by construction) and the group ϕ(_Z^N)is isomorphic toΓ.

We can define an action of Z^N−d×E(ker(h_σC)) on C^I by setting for (m,E(t))∈_Z^N−d×E(_ker(h_σC))_andz∈_C^I_,

(m,t)·z =E(ϕ(0⊕m)) +t)z

Remark4.2. 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 ofT^I onC^I =U_eσcommutes with this action. Hence, we can form the quotient[_C^I_/Z^N−d×E(_ker(h_σC))]_.

Lemma 4.3. The quotient stack [_C^I_/Z^N−d×E(ker(h_σC))]depends neither on the choice of the basisB nor on the permutationχ. Hence, the action induced by the morphism ϕdepends only onσ.

Proof. Let (B = (v_i,i ∈ eI),χ) and (B⁰ = (v_i,i ∈ eI⁰),χ⁰) be two pairs of basis and permutation. Then we can define two associated isomorphisms ψ : C^eI → _C^d and ψ⁰ : C^eI0 → _C^d and two morphisms ϕ : C^N → _C^I, ϕ⁰ : C^N → _C^I0 . Then, the identity is a α-equivariant morphism, where α is the morphismZ^N−d×E(_ker(h_σ))→_Z^N−d×E(_ker(h_σ))_{defined by}

α(m,E(y)) = (P_χ⁻0¹P_χ(m)_,E(ϕ(m)−ψ⁰⁻¹h(ϕ(m)))E(y)) where Pχ is the linear map associated toχ.

Indeed, the following diagram is commutative

E(h⁻_σ¹(_Γ)) ^{= //}E(h⁻_σ¹(_Γ))

Z^N−d×E(ker(h_σ))

action throughϕ

OO

α //Z^N−d×E(ker(h_σ))

action throughϕ⁰

OO

Definition 4.4. The stack U^cal_σ := [_C^I_/Z^N−d×E(ker(hσC))]is the quantum toric variety associated to the coneσ and to the calibrationh^cal :Z^N →_Γ. Remark 4.5. If σ is non-simplicial (i.e. ker(hσC) is non-empty and thus is not discrete) then the quantum toric variety associated to σ cannot be describe as a quasifold (i.e. locally the quotient of a spaceRⁿby a discrete subgroup, see [Pra01]) in contrast to the simplicial case.

The associated "torus" to U_σ^calis the stack[_T^I/Z^N−d×E(ker(h_σC))]. We can endow this stack with a structure of presented calibrated quantum torus thanks to the morphisms used to define U_σ^cal :

Proposition 4.6. (T_h,^cal_I,ϕ : Z^N → ψ⁻¹(_Γ),h_σC,χ⁻¹(I),P_χ : e_i 7→ e_χ(i),χ) is a presented calibrated quantum torus which encodes the stack [_T^I_/Z^N−d× E(ker(h_σC))]

Proof. This proposition comes from the fact that ψ is induced by h_σC on C^I/ ker(h_σC)'_C^eI and the equality ϕ= ψ⁻¹hP_χ. Notation 4.7. In what follows, we will omit the isomorphism and just write[_T^p_/Z^N−d×E(_ker(h_σC))]instead of this 6-uple.

Warning 4.8. In the classical non-simplicial case (i.e. Γ is a lattice and the calibration is an isomorphism), the stack U_σ^cal is not a (stack representable by a) variety.

However, if we replace the stack quotient by the GIT quotient, we get the classical toric variety associated to σ:

Proposition 4.9. If Γ is discrete, h is an isomorphism and the set of virtual generators is empty, then we can define the (classical) toric variety Uσ associated toσ(andΓ) and

Uσ=_C^I_Z^N−d×E(ker(h_σC))

where the 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_σC)) for the trivial line bundle C^I ×_C → _C^I with the linearization defined for all (m,t)∈_Z^N−d×E(_ker(h_σC))and for(z,w)∈ _C^I×_C,

(m,t)·(z,w) = ((m,t)·z,E(hϕ(0⊕m) +t,ai)w) for any a∈_Z^I.

Proof. Sincehis an isomorphism, the action of the groupZ^n−d×E(ker(h_σC)) on C^I is the same action as the action of E(h⁻_σC¹(_Γ)) on it thanks to the equality

(4.2) {E(ϕ(0⊕m) +t)|m∈_Z^N−d,E(t)∈ E(ker(h_σC))}= E(h⁻_σC¹(_Γ)).

If we suppose Γdiscrete, we can suppose too thatΓ = _Z^d without loss of generality. In [CLS11] (theorem 5.1.11), the authors prove thatC^I →U_σ is an almost geometric quotient for the action ofGdefined 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 the variety U_σ see [CLS11] p172/173).

SinceC^∗ is a divisible group, we have a exact sequence :

(4.3) 0 ^//G ^//T^{I ϕ //}T^{d //}0

where the morphism ϕis the induced morphism by Hom(M,C^∗)and the isomorphisms Hom(_Z^I,C^∗) → _T^I, (u : Z^I → _C^∗) 7→ (u(e_i),i ∈ I) and Hom(_Z^d,C^∗)→_T^d,(u:Z^d→_C^∗)7→ (u(e₁), . . . ,u(e_d)).

Then, we deduce from the equality h_σC = M_C^T that ϕis the morphism T^I → _T^d induced byh. Finally, thanks to the exact sequence (4.3), we get the isomorphismG' E(h⁻_σC¹(_Z^d))(since ker(E) =_Z^d)

The semi-stability is proved in chapter 12 of [Dol03].

We cannot have the equality

C^IZ^N−d×E(ker(h_σC)) = [_C^I_/Z^N−d×E(ker(h_σC))].

Indeed, the categorical quotient (for the action ofG)C^I →U_σis geometric if, and only if, σ is a simplicial cone because the set of semi-stable points is 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=_Ze1+_Ze2+_Ze3+_Z(ae₁−be₂+ce₃) +

∑

Zvi ⊂_R³

with a,b,c ∈ _R>0,v_i ∈ _R³ be a subgroup of R³, h^cal : Z^N → _Γ be the calibration ofΓdefined byh^cal(e_i) =e_i fori=1, 2, 3,h^cal(e₄) =ae₁−be₂+ ce₃ andh^cal(e_i) = v_i for i≥5. Let σ = Cone(e₁,e₂,e₃,ae₁−be₂+ce₃)be a strongly convex cone ofR⁴.

The morphism h_σ is the restricted map h^cal_|Z4⊕0 and the kernel ker(h_σC) is the line

ker(h_σC) =_C(−a,b,−c, 1)

We have a decompositionC⁴ = (_C³⊕0)⊕ker(_hσC)and hence, an action of Z^N−3×E(ker(h_σC))on C⁴defined by :

(4.4) (m,E(t))·s =E(h(₀⊕m) +t)s Then, U_σ^cal = [_C⁴_/Z^N−3×E(_C(−a,b,−c, 1))]

For the case N = _{4 and} a = b= c =1, thanks to the decomposition of S, we find a stacky version of the toric varietyC⁴G =V(yt−xz)⊂ _C⁴ (see [Ful93] p17)

Figure1. The coneσ

4.1.2. Cone of dimension k < d. Supposeσ = σ_I ⊂ _R^d is a cone of dimen- sion k<d.

Let J a subset of[[1,N]]of cardinald−ksuch that C^d =Vect(σ)⊕Vect(v_j,j∈ J)

Hence, h_σ : C^I ⊕_C^J → _C^d,e_i 7→ v_i is an epimorphism. Now, we can reuse the discussion of subsection 4.1.1 and define an action of Z^N−d× E(ker(h_σC)) 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_eσ×0 is preserved by this action and thus, we can define the affine quantum toric variety associated toσ:

U^cal_σ := [_C^I×_T^J/Z^N−d×ker(h_σ)]

Proposition 4.11. The affine quantum toric variety U_σ^cal is well-defined i.e. if we change the family(v_j,j∈ J), we get an isomorphism which respects a cocycle condition.

Proof. Let J and J⁰ be two subsets of [[1,N]] of cardinal d−k such that dim(_Vect(v_j,j ∈ J)) = _dim(_Vect(v_j,j⁰ ∈ J⁰)) = d−k and χ, χ⁰ be the associated permutations (and Pχ, P_χ⁰ the associated linear maps). The morphism P_χ⁰ ◦P_χ⁻¹ : C^J → _C^J0 is an isomorphism and id⊕P_χ⁰◦P_χ⁻¹ : C^I⊕_C^J →_C^I⊕_C^J0 too.

We will prove that this morphism induces an isomorphism of stacks [_C^I×_T^J/Z^N−d×ker(h_σC⊕h_J)]'[_C^I×_T^J0/Z^N−d×ker(h_σC⊕h_J⁰)]

where h_J (resp. h_J⁰) is the restriction ofh^cal_C onZ^J (resp. Z^J0)

Firstly, we can remark thatx⊕y∈_ker(h_σC⊕h_J) =_ker(h_σC⊕h_J⁰)_{if, and} only if, x ∈ ker(h_σC)andy =0. Thus, we get the following commutative diagram (every arrow is an isomorphism) :

C^I⊕_C^J/ ker(h_σC⊕h_J) ^{id⊕Pχ0◦P}

−1 χ //

C^I⊕_C^J0/ ker(h_σC⊕h_J⁰)

C^I/ ker(h_σC)⊕_C^{J //}_C^I/ ker(h_σC)⊕_C^J0

By construction, the isomorphism id⊕P_χ⁻0¹◦P_χ descends to quotient.

Hence, we get : C^I⊕_C^{J E //}

id⊕P_χ0◦P_χ⁻¹

T^I×_T^{J  //}

C^I×_T^{J //}

[_C^I×_T^J_/Z^N−d×_ker(h_σC⊕h_J)]

h_{J J}0

C^I⊕_C^{J0 E //} _T^I×_T^{J0  //}_C^I×_T^{J //}[_C^I×_T^J_/Z^N−d×ker(h_σC⊕h_J⁰)]

The rightmost morphism is the desired morphism, which is an isomor- phism.

If we take a third subset J⁰⁰ (and a permutation χ⁰⁰) , we get two other isomorphisms h_{J J}⁰⁰ andh_J⁰_J⁰⁰. The equality

(id⊕P_χ⁰⁰◦P_χ⁻0¹)◦(id⊕P_χ⁰◦P_χ⁻¹) =id⊕P_χ⁰⁰ ◦P_χ⁻¹ induces an equality between the stack (iso)morphisms

h_J⁰⁰_J⁰◦h_J⁰_J = h_J⁰⁰_J

The following proposition is proved by the same manner as proposition 4.9.

Proposition 4.12. If Γ is discrete, h is an isomorphism and the set of virtual generators is empty then

U_σ=_C^I×_T^J_Z^N−d×E ker

h_σ Example 4.13. We will describe a variant of example 4.10 :

Let

Γ=_Ze1+_Ze2+_Ze3+_Z(ae₁−be₂+ce₃) +

∑

v_iZ⊂_R⁴

be a subgroup ofR⁴,h: Z^N →_Γbe the calibration ofΓdefined byh(e_i) = e_i for i = 1, 2, 3, h(e₄) = ae₁−be2+ce3 and h(e_i) = v_i for i ≥ 5, and σ=Cone(e₁,e₂,e₃,ae₁−be₂+ce₃).

Note k ∈ {_{5, . . . ,}N}an integer such that (e₁,e₂,e₃,v_k)is a basis of C⁴. The kernel of the morphism hσ: C⁴⊕_C^{k} → _C⁴ isC(−a,b,−c, 1, 0) i.e.

ker(hσC)⊕0. Then, U_σ^cal = [_C⁴×_C^∗_/Z^N−3×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.

Proposition 4.14. Letσ = σ_I be a cone and letτ= σ_I⁰ be a face of σ. Then we have an isomorphism

U_τ^cal '[_C^I0×_T^I\I0×_T^J/Z^N−d×E(ker(h_σC))],→ U_σ^cal which restricts to a torus isomorphism

[_T^I0×_T^J0_/Z^N−d×E(ker(h_τC))]'[_T^I0×_T^I\I0×_T^J_/Z^N−d×E(ker(h_σC))]

induced by the identity of T_h,^cal_I.

Proof. Without loss of generality, we can suppose that J⁰ = JäK and C^I∪J =_C^eI⊕ker(h_σC)⊕_C^K.

Then, the map f: C^I0×_T^J0 = (_C^eI⊕ker(h_τC))×_T^J0→ (_C^eI⊕E_I\I(ker(h_σC))× T^J0 = _C^I0×_T^I\I0×_T^J (where E_I\I : (z,w) ∈ _C^I0×_C^I\I0 7→ (z,E(w)) ∈ C^I0×_T^I\I0) defined by

f(x⊕t,y) = (x,E_I\I(t, 0),z)

descends to the desired stack isomorphism

4.1.3. Toric morphisms of affine quantum toric varieties. We will use the same definition of toric morphism as [KLMV20] :

Definition 4.15. A toric morphism between the two affine quantum toric varieties U_σ^cal₁ = [_C^I×_T^J/G₁] and U_σ^cal₂ = [_C^I0×_T^J0/G2] is a stack mor- phism [_C^I×_T^J/G₁] →[_C^I0×_T^J0/G₂]which restricts to a presented cali- brated torus morphism[_T^I×_T^J/G₁]→[_T^I0×_T^J0/G₂]

By definition of torus morphism, such torus morphism induces a torus morphism between standard quantum tori T_h,^cal_I → T_h^cal0,I⁰ and thus two lin- ear maps (L : R^d → _R^d0,H : R^N → _R^N0). Moreover, L(σ) ⊂ σ⁰ and H(_bσ)⊂ _bσ⁰. 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, σ⁰ be a cone of dimension k⁰ of R^d0. Leteσ andeσ⁰ be the associated cone inR^N andR^N0.

Let (L,H):(σ,h)→(σ⁰,h⁰)be a calibrated quantum fan morphism.

Let σ = σ_I be a cone of dimension k of ∆ and eσ the associated cone. By definition of calibrated quantum fan morphism, there exists a coneeσ⁰ = _fσ_I⁰ of ∆⁰_h0such that H(_eσ)⊂_eσ⁰.

We will adapt the construction of [KLMV20] (section 5.1). Firstly, we begin by replacing the calibration h by the morphism ϕ (used in definition of

U_σ^cal) in diagram (2.2) :

Let J be a subset of cardinald−kof{1, . . . ,N}such that C^d=Vect_C(σ)⊕Vect_C(v_j,j∈ J).

Let eJ be a subset of J such that for all j ∈ eJ, L(v_j) ∈/ L(Vect_C(σ)) and such that the family (L(v_j),j ∈ eJ) is free. Let J⁰ be a subset of cardinal

d⁰−dim(σ⁰)of{1, . . . ,N}containing eJ such that C^d0 =Vect_C(σ⁰)⊕Vect_C(v⁰_j,j∈ J⁰).

Note h_J : C^J → _C^d (resp. h_J⁰ : C^J0 → _C^d0) the linear map e_j 7→ v_j for all j ∈ J (resp. e_j 7→ v⁰_j for all j ∈ J⁰) To sum up, we have the following commutative diagram :

(4.5) C^I⊕_C^{J hσC+hJ //}

eL

C^d

L

C^I0⊕_C^{J0 hσ0C+hJ0 //}_C^d0

whereeLis the mapC^I⊕_C^J =ker(h_σ)⊕(_C^eI⊕_C^J)→ker(h⁰_σ0)⊕(_C^eI0⊕_C^J0) defined by :

eL(w,z) = (H(w),ψ⁰⁻¹(L(h_σC(z))))

where ψ⁰ is the map induced by h⁰_σ0 used in the definition of U_σ^cal. This map is well defined i.e. H(ker(h_σC)) ⊂ ker(h⁰_σ0C) since H(_bσ) ⊂ _bσ⁰ and (L,H)is a morphism of calibrated quantum fans.

Letχandχ⁰ be the permutations used to defined the toric varieties U_σ^cal and U_σ^cal0 i.e. permutations such that

χ({1, . . . ,k}) =eI,χ({k+1, . . . ,d}) = J and

χ⁰({1, . . . ,k⁰}) = eI⁰,χ⁰({k⁰+1, . . . ,d⁰}) = J⁰.

Let P_χ∈ GL_N(_R)andP_χ⁰ ∈GL_N⁰(_R)be the associated linear maps.

Moreover, we can extend this diagram with the morphisms ϕ = ψ⁻¹◦ h_C◦P_χ:C^N →_C^I⊕_C^J _and ϕ⁰ = ψ⁰⁻¹◦h⁰_C◦P_χ⁰: C^N0 → _C^I0⊕_C^J0 _{used to} define the quantum toric varieties U_σ^cal and U_σ^cal0 (and more precisely, used to define an action ofZ^N−d(resp. Z^n0−d0) onC^I⊕_C^J (resp.C^I0⊕_C^J0) :

(4.6) C^{N ϕ //}

P⁻¹

χ0 HPχ

C^I⊕_C^{J hσC+hJ //}

eL

C^d

L

C^{N0 ϕ //}C^I0⊕_C^{J0 hσ0C+hJ0 //}_C^d0

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_χχ⁰ := P_χ⁻0¹HP_χ,eL)a toric morphism U_σ^cal → U_σ^cal0 :

NoteE_I :C^I×_C^J →_T^I×_C^J the map defined by : ((z_i)_i∈I,(w_j)_j∈J)7→ ((E(z_i))_i∈I,(w_j)_j∈J) andE_J :C^I×_C^J →_C^I×_T^J the map defined by :

((z_i)_i∈I,(w_j)_j∈J)7→ ((z_i)_i∈I,(E(w_j))_j∈J)