"Consider an equivariant vector bundle
E of rank
k on a toric variety
X corresponding to a fan
Σ. Then
E is
trivial on any invariant open affine subvariety of
X and moreover homogeneously generated by
k elements.
Furthermore, the transition maps between these trivializations are homogeneous of degree zero. Thus,
after fixing local homogeneous generators, we get a list of degrees of generators for each cone
in
Σ, along with a transition map for each pair of cones. Conversely, given a list of
k degrees for every
cone of
Σ along with transition maps satisfying compatibility and regularity conditions for every pair of cones,
one can construct an equivariant vector bundle of rank
k on
X."
"This description of equivariant vector bundles, due to Kaneyama, is implemented for complete, pointed fans in the following way:
It is only necessary to consider charts corresponding to maximal dimensional cones of
Σ. Furthermore, each codimension-one cone of
Σ corresponds to a pair of maximal dimensional cones, and thus to a transition map. Due to the compatibility condition for transition maps,
one can reconstruct the transition map corresponding to an arbitrary pair from the maps of this sort. If the dimension of
Σ is
n then
for each maximal dimensional cone the degree list of the corresponding chart is saved as an
n times
k matrix over ",TO ZZ,", giving
k degree vectors in the dual lattice of the fan, one for each local generator of the bundle. Additionally,
for every pair of maximal cones intersecting in a common codimension-one face, there is a matrix in
GL(
k,",TO QQ,"), representing the transition map between these two affine charts. Indeed, suppose that
cones
σ1 and
σ1 intersect in some codimension-one face, with corresponding affine
charts
U1 and
U2. Then on the intersection, the
i-th generator for
U1 has a unique
representation as a linear combination in the generators for
U2 after being multiplied with characters to all
have the required degree. The coefficients in this representation form the
i-th column of the desired matrix."
"We briefly consider the example of
ℙ2, corresponding to the complete fan with rays
through
(0,1),
(1,0), and
(-1,-1). Denote by
x the character of weight
[1,0] and by
y the character
of weight
[0,1]. Now the coordinate rings of the three standard affine charts of
ℙ2 are generated by
respectively
(x -1,x -1y),
(x,y), and
(xy -1,y -1). This means that the modules of differentials
are generated by respectively
(d(x -1),d(x -1y)),
(dx,dy), and
(d(xy -1),d(y -1)). These modules give us
local trivializations of the cotangent bundle on
ℙ2. The degrees of the generators for the first chart
then are
[-1,0] and
[-1,1], for example. Now, since
d(x -1)=-x -2dx and
d(x -1y) = -x -2ydx + x -1dy,
we get that the transition map between the generators of the first and second chart is given by the matrix with
columns
(-1,0) and
(-1,1)."
An instance of class ToricVectorBundleKlyachko, when displayed or printed, gives an overview of the
characteristics of the bundle:
i1 : E = cotangentBundle(projectiveSpaceFan 2,"Type" => "Kaneyama")
o1 = {dimension of the variety => 2 }
number of affine charts => 3
rank of the vector bundle => 2
o1 : ToricVectorBundleKaneyama
|
To see all relevant details of a bundle use
details. The data described above is all stored in a single hash table. In the example from above, the first chart has the key 0, and transition map described above has key (0,1):
i2 : details E
o2 = (HashTable{0 => (| -1 0 |, | -1 -1 |)}, HashTable{(0, 1) => | -1 -1 |})
| -1 1 | | 0 1 | | 0 1 |
1 => (| 1 0 |, | 1 0 |) (0, 2) => | 1 0 |
| 0 1 | | 0 1 | | -1 -1 |
2 => (| 1 -1 |, | 0 1 |) (1, 2) => | -1 -1 |
| 0 -1 | | -1 -1 | | 1 0 |
o2 : Sequence
|