The cohomology ring of toric orbifolds with integer coefficients
The cohomology ring of toric orbifolds with
integer coefficients
Jongbaek Song (KAIST)
(jointly with Anthony Bahri and Nigel Ray)
January 23rd, 2014
Toric Topology 2014 in Osaka
Osaka City University, Osaka, Japan
1/ 38
The cohomology ring of toric orbifolds with integer coefficients
Outline
1
Toric orbifolds
2
Equivariant cohomology and Piecewise polynomials
3
Cohomology ring and Integrality condition
4
J-construction of toric orbifolds and cohomology ring
2/ 38
The cohomology ring of toric orbifolds with integer coefficients
Toric orbifolds
1
Toric orbifolds
2
Equivariant cohomology and Piecewise polynomials
3
Cohomology ring and Integrality condition
4
J-construction of toric orbifolds and cohomology ring
3/ 38
The cohomology ring of toric orbifolds with integer coefficients
Toric orbifolds
Toric variety
A toric variety X : A normal complex algebraic variety with
algebraic (C∗ )n -action having a dense orbit.
Theorem (Fundamental theorem for toric varieties)
The category of toric varieties is equivalent to the category of
fans.
XΣ ←→ ΣX
4/ 38
The cohomology ring of toric orbifolds with integer coefficients
Toric orbifolds
Construction of toric varieties
Σ = (K, Λ) : a simplicial fan with Σ(1) = {λ1 , . . . , λm } ⊂ Zn ,
1
K : underlying simplicial complex,
[
ZΣ (C, C∗ ) =
(C|I| × (C∗ )m−|I| ),
σ∈K
2
∗ m
with standard action
m of (Cn )
Λ = λ1 | . . . |λm : Z → Z
0
/ ker ΛC 
/ (C∗ )m
5/ 38
ΛC
/ (C∗ )n
/0.
The cohomology ring of toric orbifolds with integer coefficients
Toric orbifolds
Construction of toric varieties
Σ = (K, Λ) : a simplicial fan with Σ(1) = {λ1 , . . . , λm } ⊂ Zn ,
1
K : underlying simplicial complex,
[
ZΣ (C, C∗ ) =
(C|I| × (C∗ )m−|I| ),
σ∈K
2
∗ m
with standard action
m of (Cn )
Λ = λ1 | . . . |λm : Z → Z
0
/ ker ΛC 
/ (C∗ )m
ΛC
/ (C∗ )n
/0.
XΣ = ZΣ / ker ΛC with the residual (C∗ )m / ker ΛC -action.
5/ 38
The cohomology ring of toric orbifolds with integer coefficients
Toric orbifolds
Cohomology ring of a toric variety
Correspondence...
XΣ : compact iff Σ : complete,
XΣ : a smooth variety iff Σ : smooth,
XΣ : an orbifold1 or simplicial iff Σ : simplicial.
1
has only finite quotient singularities
6/ 38
The cohomology ring of toric orbifolds with integer coefficients
Toric orbifolds
Cohomology ring of a toric variety
Correspondence...
XΣ : compact iff Σ : complete,
XΣ : a smooth variety iff Σ : smooth,
XΣ : an orbifold1 or simplicial iff Σ : simplicial.
H ∗ (XΣ ; −)
Q-coefficients
smooth variety
H ∗ (XΣ ; Q) ∼
=
Q[Σ]/JΣ
orbifold
1
has only finite quotient singularities
6/ 38
Z-coefficients
H ∗ (XΣ ; Z) ∼
=
Z[Σ]/JΣ
H ∗ (XΣ ; Z) ∼
=
(??)
Z[Σ]/JΣ
The cohomology ring of toric orbifolds with integer coefficients
Equivariant cohomology and Piecewise polynomials
1
Toric orbifolds
2
Equivariant cohomology and Piecewise polynomials
3
Cohomology ring and Integrality condition
4
J-construction of toric orbifolds and cohomology ring
7/ 38
The cohomology ring of toric orbifolds with integer coefficients
Equivariant cohomology and Piecewise polynomials
Piecewise polynomials
Definition
Let Σ be a fan in N. A function
[
f : |Σ| =
σ ⊂ NR → Z
σ∈Σ
is called piecewise polynomial on Σ, if it coincides with some
globally defined polynomial on each cone σ ∈ Σ
8/ 38
The cohomology ring of toric orbifolds with integer coefficients
Equivariant cohomology and Piecewise polynomials
Piecewise polynomials
Definition
Let Σ be a fan in N. A function
[
f : |Σ| =
σ ⊂ NR → Z
σ∈Σ
is called piecewise polynomial on Σ, if it coincides with some
globally defined polynomial on each cone σ ∈ Σ
PP[Σ] := PP[Σ; Z] : set of piecewise polynomials on Σ.
Z[N] : the ring of global polynomial functions on N.
{fσ } := {fσ ∈ Z[N] | σ ∈ Σ(n) } : an element in PP[Σ].
PP[Σ] has Z[N](∼
= H ∗ (BT))-algebra structure.
8/ 38
The cohomology ring of toric orbifolds with integer coefficients
Equivariant cohomology and Piecewise polynomials
Example
λ2 = (0, 1)
Σ=
∆2 ,
1 0 −1
0 1 −1
!
σ1
σ0
λ1 = (1, 0)
σ2
λ0 = (−1, −1)


fσ0 − fσ1 ∈ hxi


PP[Σ] = (fσ0 , fσ1 , fσ2 ) ∈ Z[x, y]3 fσ1 − fσ2 ∈ hy − xi
.


fσ − fσ ∈ hyi
2
0
9/ 38
The cohomology ring of toric orbifolds with integer coefficients
Equivariant cohomology and Piecewise polynomials
Proposition (A.Bahri, M.Franz, and N.Ray, 2009)
Σ : a polytopal fan in N
XΣ : associated compact projective toric variety with
H odd (XΣ ; Z) = 0.
⇒ HT∗ (XΣ ; Z) ∼
= PP[Σ] as H ∗ (BT)-algebras.
10/ 38
The cohomology ring of toric orbifolds with integer coefficients
Cohomology ring and Integrality condition
1
Toric orbifolds
2
Equivariant cohomology and Piecewise polynomials
3
Cohomology ring and Integrality condition
4
J-construction of toric orbifolds and cohomology ring
11/ 38
The cohomology ring of toric orbifolds with integer coefficients
Cohomology ring and Integrality condition
From Piecewise polynomials to Stanley-Reisner ring
Σ : a polytopal fan in Zn with Σ(1) = {λ1 , . . . , λm }.
Λ : Zm → Zn , ei 7→ λi .
b = {Λ−1 (σ) | σ ∈ Σ} : fan in Zm .
Define Σ
σ = cone(λi1 , . . . , λin ) ∈ Σ ⇒
b
σ̂ := Λ−1 (σ) = cone{ei1 , . . . , ein } ∈ Σ,
b : the ring of piecewise polynomials on Σ,
b whose
PP[Σ]
elements are of the following form
b (n) }
{f̂σ̂ } = {f̂σ̂ (xi1 , . . . , xin ) | σ̂ = cone(ei1 , . . . , ein ) ∈ Σ
12/ 38
The cohomology ring of toric orbifolds with integer coefficients
Cohomology ring and Integrality condition
Example



1 0 0
1 1 −1
b = ∆2 , 0 1 0
Σ = ∆2 ,
=⇒ Σ
0 5 −3
0 0 1
v
x3
λ2 = (1, 5)
fσ3 (u, v)
fσ1 (u, v)
u
λ1 = (1, 0)
f̂σˆ1 (x2 , x3 )
=⇒
f̂σˆ2 (x1 , x3 )
x2
fσ2 (u, v)
λ3 = (−1, −3)
x1
Σ
f̂σˆ3 (x1 , x2 )
b
Σ
13/ 38
The cohomology ring of toric orbifolds with integer coefficients
Cohomology ring and Integrality condition
Given a fan Σ, we have 3 algebraic objects...
(1)
Z[Σ] o
b
/ PP[Σ]
;
c
(3)
#
{
(2)
PP[Σ]
RELATION??
14/ 38
The cohomology ring of toric orbifolds with integer coefficients
Cohomology ring and Integrality condition
Lemma (1)
For a polytopal fan Σ with |Σ(1) | = m,
b and
Z[Σ] = Z[x1 , . . . , xm ]/IΣ ∼
= PP[Σ],
b Q],
Q[Σ] = Q[x1 , . . . , xm ]/IΣ ∼
= PP[Σ;
where IΣ is the Stanley-Reisner ideal of Σ.
15/ 38
The cohomology ring of toric orbifolds with integer coefficients
Cohomology ring and Integrality condition
Lemma (1)
For a polytopal fan Σ with |Σ(1) | = m,
b and
Z[Σ] = Z[x1 , . . . , xm ]/IΣ ∼
= PP[Σ],
b Q],
Q[Σ] = Q[x1 , . . . , xm ]/IΣ ∼
= PP[Σ;
where IΣ is the Stanley-Reisner ideal of Σ.
Sketch of proof.
b restriction to each cone of Σ.
b
α : Z[x1 , . . . , xm ] → PP[Σ],
=⇒ surjective ring homomorphism.
Sujectivity follows from inclusion and exclusion principle.
ker α = IΣ .
15/ 38
The cohomology ring of toric orbifolds with integer coefficients
Cohomology ring and Integrality condition
Lemma (2)
b Q].
PP[Σ; Q] ∼
= PP[Σ;
16/ 38
The cohomology ring of toric orbifolds with integer coefficients
Cohomology ring and Integrality condition
Lemma (2)
b Q].
PP[Σ; Q] ∼
= PP[Σ;
Sketch of proof.
b Q] o
PP[Σ;
β
/ PP[Σ; Q]
γ
T
(n)
β({f̂σ̂ }) = {f̂σ̂ (Λ−1
σ · (u1 , . . . , un ) ) | σ ∈ Σ },
b (n) }.
γ({fσ }) = {fσ (Λσ · xT ) | σ̂ ∈ Σ
σ
=⇒ β ◦ γ = idPP[Σ;Q] and γ ◦ β = idPP[Σ]
b .
16/ 38
The cohomology ring of toric orbifolds with integer coefficients
Cohomology ring and Integrality condition
Corollary
Q[Σ] ∼
= PP[Σ; Q].
17/ 38
The cohomology ring of toric orbifolds with integer coefficients
Cohomology ring and Integrality condition
Corollary
Q[Σ] ∼
= PP[Σ; Q].
Finally, in Q-coefficients,
α
∼
=
Q[Σ]
β◦α
$
b Q]
/ PP[Σ;
=
β
~
PP[Σ; Q]
17/ 38
∼
=
γ
The cohomology ring of toric orbifolds with integer coefficients
Cohomology ring and Integrality condition
Integrality condition
In order to work with Z-coefficients, we need the following
definitions.
Definition
We say a piecewise polynomial
b (n) } ∈ PP[Σ]
b
{f̂σ̂ } = {f̂σ̂ (xi1 , . . . , xin ) | σ̂ = cone(ei1 , . . . , ein ) ∈ Σ
satisfies integrality condition associated to Σ, if
T
β({f̂σ̂ }) = {f̂σ̂ (Λ−1
σ · (u1 , . . . , un ) )}
b (n) .
has integer coefficients, for all σ̂ ∈ Σ
18/ 38
The cohomology ring of toric orbifolds with integer coefficients
Cohomology ring and Integrality condition
Definition
We say an element [h(x1 , . . . , xm )] ∈ Z[Σ] satisfies integrality
condition associated to Σ, if
T
β ◦ α([h]) = {h(iσ (Λ−1
σ (u1 , . . . , un ) ))}
has integer coefficients, for all σ ∈ Σ(n)
19/ 38
The cohomology ring of toric orbifolds with integer coefficients
Cohomology ring and Integrality condition
Proposition
b (resp. α−1 ◦ γ : PP[Σ] → Z[Σ])
The image of γ : PP[Σ] → PP[Σ]
is the collection of elements in PP[Σ] (resp. Z[Σ]) which satisfy
the integrality condition associated to Σ.
Notations...
b Z = im(γ : PP[Σ] → PP[Σ])
b
PP[Σ]
Z[Σ]Z = im(α−1 ◦ γ : PP[Σ] → Z[Σ])
20/ 38
The cohomology ring of toric orbifolds with integer coefficients
Cohomology ring and Integrality condition
Finally, we have the following commutative diagram :
α0
∼
=
Z[Σ]Z
∼
=
β 0 ◦α0
%
bZ,
/ PP[Σ]
>
β0
~
∼
=
γ
PP[Σ]
HT∗ (XΣ ; Z)
where α0 and β 0 is the restrictions to their subrings satisfying
integrality conditions.
21/ 38
The cohomology ring of toric orbifolds with integer coefficients
Cohomology ring and Integrality condition
Theorem
Σ : a polytopal fan in N
XΣ : associated compact projective toric variety with
H odd (XΣ ; Z) = 0.
=⇒ H ∗ (XΣ ) ∼
= Z[Σ]Z /(JΣ )Z ,
where JΣ is the ideal generated by linear relations determined
by Σ and (JΣ )Z = JΣ ∩ Z[Σ]Z .
22/ 38
The cohomology ring of toric orbifolds with integer coefficients
Cohomology ring and Integrality condition
Theorem
Σ : a polytopal fan in N
XΣ : associated compact projective toric variety with
H odd (XΣ ; Z) = 0.
=⇒ H ∗ (XΣ ) ∼
= Z[Σ]Z /(JΣ )Z ,
where JΣ is the ideal generated by linear relations determined
by Σ and (JΣ )Z = JΣ ∩ Z[Σ]Z .
Sketch of proof.
H odd (XΣ ; Z) = 0 ⇒ Serre spectral sequence collapses, which
gives the isomorphism
H ∗ (XΣ ; Z) ∼
= HT∗ (XΣ ; Z) ⊗H∗(BT) Z
∼
= H ∗ (XΣ ; Z)/JΣ ∼
= Z[Σ]Z /JΣ
T
22/ 38
The cohomology ring of toric orbifolds with integer coefficients
Cohomology ring and Integrality condition
Finally, we have...
H ∗ (XΣ ; −)
Q-coefficients
smooth variety
H ∗ (XΣ ; Q) ∼
=
Q[Σ]/JΣ
orbifold
23/ 38
Z-coefficients
H ∗ (XΣ ; Z) ∼
=
Z[Σ]/JΣ
H ∗ (XΣ ; Z) ∼
=
Z[Σ]Z /(JΣ )Z
The cohomology ring of toric orbifolds with integer coefficients
Cohomology ring and Integrality condition
Comprehension of Integrality condition
Recall the diagram and maps
α0
∼
=
Z[Σ]Z
∼
=
β 0 ◦α0
#
bZ,
/ PP[Σ]
A
β0
∼
=
γ
PP[Σ]
T
β 0 ({f̂σ̂ }) = {f̂σ̂ (Λ−1
σ · (u1 , . . . , un ) ) | σ ∈ Σ},
T
α0 ◦ β 0 ([h]) = {h(iσ (Λ−1
σ (u1 , . . . , un ) )) | σ ∈ Σ}.
24/ 38
The cohomology ring of toric orbifolds with integer coefficients
Cohomology ring and Integrality condition
Example (XΣ ∼
= CP2(2,3,5) )
λ2 = (1, 5)
Σ:
λ1 = (1, 0)
λ3 = (−1, −3)
1
5
1
Λ13 =
0
1
Λ12 =
0
Λ23 =
−1
−3
Λ−1
23 =
− 32
− 52
−1
−3
Λ−1
13 =
1
0
1
5
Λ−1
12 =
1
0
!
1
2
,
1
2!
− 13
,
− 13
!
− 15
.
1
5
∴ HT∗ (CP2(2,3,5) ; Z) ∼
= PP[Σ] ∼
= Z[Σ]Z =
(Z[x1 , x2 , x3 ]/(x1 x2 x3 ))Z =


3
1
5
1

h 0, − 2 1u1 + 2 u21, − 2u1 + 2 u2 ∈ Z[u1 , u2 ] 
[h] ∈ Z[Σ] h u1 − 3 u2 , 0, − 3 u2 ∈ Z[u1 , u2 ]
.


h u1 − 1 u2 , 1 u2 , 0 ∈ Z[u1 , u2 ]
5
5
25/ 38
The cohomology ring of toric orbifolds with integer coefficients
Cohomology ring and Integrality condition
H ∗ (CP2(2,3,5) ; Z) and Integrality matrix


h 0, − 3 u1 + 1 u2 , − 5 u1 + 1 u2 ∈ Z[u1 , u2 ] 

2
2
2
2
Z[Σ]Z = [h] ∈ Z[Σ] h u1 − 31 u2 , 0, − 31 u2 ∈ Z[u1 , u2 ]
.


h u1 − 1 u2 , 1 u2 , 0 ∈ Z[u1 , u2 ]
5
5
“Integrality matrix”
x
 1
0

3
GΣ =
5
x2
2
0
5
x3

2
! cone{λ2 , λ3 }

! cone{λ1 , λ3 }
3
! cone{λ1 , λ2 }
0
degree 2 monomials : 15x1 , 10x2 , 6x3 .
degree 4 monomials :
225x12 , 100x22 , 36x32 , 4x2 x3 , 9x26/
1 x38
3 , 25x1 x2 .
The cohomology ring of toric orbifolds with integer coefficients
Cohomology ring and Integrality condition
1 1 −1
2
Σ= ∆ ,
−→ JΣ = (x1 + x2 − x3 , 5x2 − 3x3 )
0 5 −3
Hence,
degree 2 monomials : 15x1 = 10x2 = 6x3 =: w1
degree 4 monomials :
1 2
1 2
w1 , 9x1 x3 = 10
w1 , 25x1 x2 = 16 w21 .
4x2 x3 = 15
Finally, choose degree 4 generator
w2 := 9x1 x3 − 4x2 x3 = w2 =
1 2
w .
30 1
H ∗ (CP2(2,3,5) ; Z) ∼
= Z[Σ]Z /(JΣ )Z
∼
= Z[w1 , w2 ]/ < w21 − 30w2 , w1 w2 >
27/ 38
The cohomology ring of toric orbifolds with integer coefficients
J-construction of toric orbifolds and cohomology ring
1
Toric orbifolds
2
Equivariant cohomology and Piecewise polynomials
3
Cohomology ring and Integrality condition
4
J-construction of toric orbifolds and cohomology ring
28/ 38
The cohomology ring of toric orbifolds with integer coefficients
J-construction of toric orbifolds and cohomology ring
J-construction and canonical extension
Σ = (K, Λ) : a fan in Zn with Σ(1) = {λ1 , . . . , λm },
1
2
K : underlying simplicial complex on V = {v1 , . . . , vm } with
minimal
non-faces
{vi1 , . . . , vi` }
Λ = λ1 | · · · | λm
J = (j1 , . . . , jm ) ∈ Nm
29/ 38
The cohomology ring of toric orbifolds with integer coefficients
J-construction of toric orbifolds and cohomology ring
J-construction and canonical extension
Σ = (K, Λ) : a fan in Zn with Σ(1) = {λ1 , . . . , λm },
1
2
K : underlying simplicial complex on V = {v1 , . . . , vm } with
minimal
non-faces
{vi1 , . . . , vi` }
Λ = λ1 | · · · | λm
J = (j1 , . . . , jm ) ∈ Nm
Σ(J) = (K(J), Λ(J)) : a fan in Zn−m+
1
Pm
i=1 jj
with
K(J) : simplicial complex on
{v11 , . . . , vj11 , . . . , v1i , . . . , vjii , . . . , v1m , . . . , vjmm }
| {z }
| {z }
| {z }
ji
j1
2
jm
with minimal non-faces of the form Vi1 ∪ · · · ∪ Vi` , where
Vi = {v1i , . . . , vjii }.
Λ(J) = .....
29/ 38
The cohomology ring of toric orbifolds with integer coefficients
J-construction of toric orbifolds and cohomology ring
−1
.
.
.
−1









Λ(J) = 








Ij0 −1

−1
.
.
.
−1
Ij1 −1
..
.
.
.
.
−1
.
.
.
−1
Ijn −1
Λ
30/ 38








,








The cohomology ring of toric orbifolds with integer coefficients
J-construction of toric orbifolds and cohomology ring
simplicial wedge construction
Especially, for J = (1, . . . , 1, 2 , 1, . . . 1),
↑
i−th
Σ(J) =: Σ̃ = (wedgei (K), Λ̃), “canonical extension of Σ”, where
wedgei (K) := [{i0 , i} ∗ linkK {i}] ∪ [{{i0 }, {i}} ∗ (K \ {i})],
Call it “simplicial wegde construction”.


1 0 · · · −1 · · · 0
 0



Λ̃ := Λ(J) =  .

.
 . λ1 · · · λi · · · λm 
0
31/ 38
The cohomology ring of toric orbifolds with integer coefficients
J-construction of toric orbifolds and cohomology ring
Examples
wedgei (K) := [{i0 , i} ∗ linkK {i}] ∪ [{{i0 }, {i}} ∗ (K \ {i})]
∂∆2 :
3
3
1’
1
: wedge1 (∂∆2 ) = ∂∆3
2
2
1
The cohomology ring of toric orbifolds with integer coefficients
J-construction of toric orbifolds and cohomology ring
Examples
wedgei (K) := [{i0 , i} ∗ linkK {i}] ∪ [{{i0 }, {i}} ∗ (K \ {i})]
∂∆2 :
3
3
1’
1
: wedge1 (∂∆2 ) = ∂∆3
2
2
1
2
1
2
1
∂∆ ∗ ∂∆ :
1’
3
1
3
1
4
4
32/ 38
: wedge1 (∂∆1 ∗ ∂∆1 )
= ∂∆2 ∗ ∂∆1
The cohomology ring of toric orbifolds with integer coefficients
J-construction of toric orbifolds and cohomology ring
Example (J = (1, 2, 1))
2 1 1 −1
Σ= ∆ ,
,
0 5 −3
XΣ ∼
= CP2(2,3,5)
J = (1, 2, 1)



1 0 −1 0
Σ̃ = ∆3 ,  0 1 1 −1  ,
0 0 5 −1
33/ 38
XΣ̃ ∼
= CP3(3,2,3,5)
The cohomology ring of toric orbifolds with integer coefficients
J-construction of toric orbifolds and cohomology ring
/ XΣ
Σ = (K, Λ)
Σ̃ = wedgei K, Λ̃
%
GΣ
/ H ∗ (XΣ ; Z)
??
/X
Σ̃
9 GΣ̃
/ H ∗ (X ; Z)
Σ̃
If we know (??), H ∗ (XΣ̃ ; Z) can be easily read off from the
original one.
34/ 38
The cohomology ring of toric orbifolds with integer coefficients
J-construction of toric orbifolds and cohomology ring
Proposition
Consider J = (1, . . . , 1, 2 , 1, . . . 1).
↑
i−th
Let (gσ1 , . . . , gσi , . . . , gσm ) be a row vector in GΣ corresponding to
σ ∈ KΣ .
If gσi 6= 0 (⇔ i ∈ σ), then
(gσ1 , . . . , gσi , . . . , gσm )
(ggσi , gσ1 , . . . , gσi , . . . , gσm ) in GΣ̃
If gσi = 0 (⇔ i ∈
/ σ), then
(gσ1 , . . . ,
0
↑
i−th
, . . . , gσm )
1 gσ1 . . .
0 gσ1 . . .
35/ 38
0 . . . , gσm
1 . . . , gσm
in GΣ̃
The cohomology ring of toric orbifolds with integer coefficients
J-construction of toric orbifolds and cohomology ring
Example
Example
1 1 −1
2
Recall, for Σ = ∆ ,
, and J = (1, 2, 1),
0 5 −3
XΣ ∼
= CP2(2,3,5)


0 2 2
GΣ = 3 0 3
5 5 0
XΣ̃ ∼
= CP3(3,2,3,5)

2 0 2
1 3 0
GΣ̃ = 
0 3 1
5 5 5
36/ 38

2
3

3
0
The cohomology ring of toric orbifolds with integer coefficients
J-construction of toric orbifolds and cohomology ring
Computation
Degree 2 : 10x22 , 15x1 , 10x21 , 6x3 .
By the ideal JΣ(J) , 10x22 = 15x1 = 10x21 = 6x3 =: w1 .
Degree 4 : 75x21 x1 = 75x1 x22 = 12 w21 , 100x21 x22 = w21 ,
1 2
12x21 x3 = x22 x3 = 15 x12 , 9x1 x3 = 10
w1 .
Choose 9x1 x3 =: w2 .
1 3
1
Degree 6 : 8x21 x22 x3 = 75
w1 , 9x21 x1 x3 = 9x22 x1 x3 = 100
w31 ,
1 3
125x21 x1 x23 = 12
w1 .
1
Choose 8x21 x22 x3 − x21 x1 x3 = 300
w31 =: w3 .
∴ H ∗ (CP3(3,2,3,5) ) ∼
= H ∗ (CP3(2,3,3,5) )
Z[w1 , w2 , w3 ]/ < w21 − 10w2 , w31 − 300w3 , w22 , w1 w3 > .
37/ 38
The cohomology ring of toric orbifolds with integer coefficients
J-construction of toric orbifolds and cohomology ring
Thank you for your
attention!
38/ 38