Description
"A toric vector bundle on a toric variety
X is a locally free sheaf
E together with an action of the torus
T on the geometric vector bundle
V(E) such that the projection to the base
X is equivariant, and the action of
T on the fibers is linear. There also is an induced action of
T on the local sections
s ∈ Γ(U,E) given by
(t*s)(x) = t -1(s(t x)) . This implies that a regular section
xu ∈ Γ(X,OX) for an element
u in the character lattice
M also has weight
u. Other choices for the induced action are possible. In fact, the upper one is different from Klyachko’s in his original desription where
xu ∈ Γ(X,OX) has weight
-u. We denote by
E0 the fiber over the unit
t0 ∈ T, and by
Uσ ⊂X the open affine torus invariant subset associated with the cone
σ. The primitive generator of the ray
ρ in the fan
Σ is denoted by
vρ. Evaluating local homogeneous sections
Γ(Uρ,E)u of weight
u at
t0 provides us with an embedding of these finite dimensional vector spaces into
E0. One can show that the upper choice of the induced torus action implies that the image of
Γ(Uρ,E)u1 is contained in the image of
Γ(Uρ,E)u2 if and only if the pairing
(u1-u2,vρ) ≤0. Furthermore one observes that the image only depends on the class of the weight
u in the quotient lattice
Mρ := M/Mρ, where
Mρ denotes the intersection of
M with the vector space perpendicular to the ray
ρ. Since
Mρ ≅ℤ we denote the image of
Γ(Uρ,E)u in
E0 by
Eρ(i) with
i = (u,vρ). Each ray
ρ∈ Σ thus gives rise to an increasing filtration
{Eρ(i)} of
E0. Since
E0 is finite dimensional there is only a finite set of integers
i for which a jump occurs, i.e.,
Eρ(i) strictly contains
Eρ(i-1). At all other steps the filtration remains constant. Apart from that, each open affine subset
Uσ for
σ∈ Σ induces a direct sum decomposition of
E0 = ⊕u ∈ MσEσu such that
Eρ(i) = ∑(u,vρ) ≤i Eσu for each
ρ∈ σ and
i ∈ ℤ. Observe that the lattice
Mσ is defined analogously to the lattice
Mρ, i.e., it is the quotient lattice
M/Mσ where
Mσ denotes the intersection of
M with the vector space perpendicular to the cone
σ."
With the notation and conventions introduced above it is now possible to state the fundamental theorem of Klyachko which completely describes toric vector bundles in linear algebraic terms:
The category of toric vector bundles on the toric variety X is equivalent to the category of finite dimensional k-vector spaces E0 with collections of increasing filtrations {Eρ(i)| i ∈ ℤ}, indexed by the rays of Σ, satisfying the following compatibility condition: For each cone σ∈ Σ there is a decomposition E0 = ⊕u ∈ Mσ Eu such that Eρ(i) = ∑(u,vρ) ≤i Eu for every ray ρ∈ σ and every i ∈ ℤ.
"In contrast to the implementation of Kaneyama’s description this one works for every toric variety
X i.e., there are no restrictions on the fan
Σ. For each ray
ρ of the fan
Σ there are two matrices comprising the necessary filtration data. The first one is an invertible matrix
A(ρ) ∈ GL("
k,
QQ") whose columns contain a basis of the vector space
E0 which is associated to the filtration corresponding to the ray
ρ. The second one is a ",TT "1 x k"," integer matrix, the so called filtration matrix. It determines at which step an element of the basis given in the first matrix actually contributes to a certain subspace in the filtration, i.e., if the j-th entry of the filtration matrix is i then the j-th basis vector appears at the i-th step in the filtration. Hence
Eρ(i) is generated by all basis vectors listed in
A(ρ) whose corresponding entry in the filtration matrix is less or equal to
E0."
"To link up to the description of Kaneyama we will also discuss the example of the cotangent bundle
ΩX of
X = ℙ2. Recall that
X can be given by the complete fan with rays
ρ1 = (1,0),
ρ2 = (0,1), and
ρ3 = (-1,-1). There are three maximal cones, namely
σ1 spanned by
ρ1,ρ2,
σ2 spanned by
ρ2,ρ3, and
σ3 spanned by
ρ3,ρ1. Each of them corresponds to a torus invariant affine chart
Uσi. It follows that the
k[σ1v ∩M]-module
Γ(Uσ1,ΩX) is generated by
dx := d(x[1,0]), and
dy := d(x[0,1]), and analogously for the remaining charts. We now fix a basis of
Ω0 by evaluating the sections
dx,dy at the unit
t0. This gives rise to filtrations
Ωρ(i). We only consider the example
ρ= ρ3. The filtrations for the two other rays can be found by analogous calculations. Now,
k[Uρ3] = k[x-1,x-1y,xy-1]. Then,
Γ(Uρ3,ΩX) is generated as a
k[Uρ3]-module by
-x-2dx, -x-2ydx + x-1dy. Thus,
Γ(Uρ3,ΩX)[1,0] = 0,
Γ(Uρ3,ΩX)[0,0] is generated by
xy-1(-x-2ydx + x-1dy), and
Γ(Uρ3,ΩX)[-1,0] is two-dimensional. Since
[1,0], [0,0], and
[-1,0] pair with
vρ3=(-1,-1) to respectively
-1, 0, and
1, the filtration
Ωρ3(i) jumps at
1 and
0 with corresponding basis vectors
(0,-1) and
(-1,1). Since
ΩX already is a vector bundle we do not have to check the compatibility conditions."
An instance of class ToricVectorBundleKlyachko, when displayed or printed, gives an overview of the characteristics of the bundle:
i1 : E = cotangentBundle(projectiveSpaceFan 2)
o1 = {dimension of the variety => 2 }
number of affine charts => 3
number of rays => 3
rank of the vector bundle => 2
o1 : ToricVectorBundleKlyachko
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To see all relevant details of a bundle use
details. The data described above are stored in a single hash table. In the example from above, the keys are the rays of the fan, and each of them comes with a base matrix and a filtration matrix:
i2 : details E
o2 = HashTable{| -1 | => (| 0 -1 |, | 1 0 |)}
| -1 | | -1 1 |
| 0 | => (| 0 1 |, | 1 0 |)
| 1 | | 1 0 |
| 1 | => (| 1 0 |, | 1 0 |)
| 0 | | 0 1 |
o2 : HashTable
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