The list
L must have one entry for each maximal cone
σ in the
underlying fan
Σ of
E. If the rank of the bundle is
k and
the ambient dimension of the variety is
n then each entry must either be
an
n by
k matrix over
ZZ or a list of these. Then it checks for
each maximal cone in the fan (given in the order of
maxCones(ToricVectorBundle)) if
for any of the matrices in the corresponding entry in
L these weight vectors admit a decomposition
of the bundle into torus eigenspaces. See
Sam Payne's Moduli of toric vector bundles,
Compositio Math. 144, 2008. Lemma 3.5.
One can for example use the output of the function
findWeights.
i1 : E = tangentBundle projectiveSpaceFan 3
o1 = {dimension of the variety => 3 }
number of affine charts => 4
number of rays => 4
rank of the vector bundle => 3
o1 : ToricVectorBundleKlyachko
|
i2 : L = findWeights E
o2 = {{| 1 1 1 |, | 1 1 1 |}, {| -1 0 0 |, | -1 0 0 |}, {| -1 0 0 |,
| -1 0 0 | | -1 0 0 | | 1 1 1 | | 1 1 1 | | 0 -1 0 |
| 0 0 -1 | | 0 -1 0 | | 0 0 -1 | | 0 -1 0 | | 0 0 -1 |
------------------------------------------------------------------------
| -1 0 0 |}, {| -1 0 0 |, | -1 0 0 |}}
| 0 0 -1 | | 0 -1 0 | | 0 0 -1 |
| 0 -1 0 | | 1 1 1 | | 1 1 1 |
o2 : List
|
i3 : existsDecomposition(E,L)
o3 = true
|
Note that the data given in the description of
E defines an equivariant vector bundle
on the toric variety exactly if there exists a set of weight vectors for each maximal cone that admits a
decomposition. The function
isVectorBundle uses this.