existsDecomposition -- checks if a list of matrices of weight vectors for each maximal cone admits a decomposition

Synopsis

• Usage:

  b = existsDecomposition(E,L)

• Inputs:
  • E, a vector bundle on a toric variety (Klyachko's description)
  • L, a list
• Outputs:
  • b, a Boolean value, whether there exists a decomposition

Description

The list L must have one entry for each maximal cone $\sigma$ in the underlying fan $\Sigma$ 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 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  0 -1 |  | 0  -1 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.

See also

• findWeights -- finds the possible weight vectors for the maximal cones
• isVectorBundle -- checks if the data does in fact define an equivariant toric vector bundle
• maxCones(ToricVectorBundle) -- the list of maximal cones of the underlying fan