Description
A toric vector bundle in Klyachko's description as used in
ToricVectorBundles is given by ascending filtrations. Unfortunately, the filtration steps are not always stored in an ascending way by certain methods, like
dual(ToricVectorBundle) or
tensor(ToricVectorBundle,ToricVectorBundle) (and maybe others). This may cause problems in some methods of
PositivityToricBundles, for example in
toricChernCharacter. This method ensures that the filtration steps are stored in an ascending way and therefore, that the methods of this package can be used safely.
i1 : T = tangentBundle 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
|
i2 : E = T ** (dual T)
o2 = {dimension of the variety => 2 }
number of affine charts => 3
number of rays => 3
rank of the vector bundle => 4
o2 : ToricVectorBundleKlyachko
|
i3 : details E
o3 = HashTable{| -1 | => (| 0 1 0 1 |, | 0 -1 1 0 |)}
| -1 | | 1 -1 1 -1 |
| 0 1 0 0 |
| 1 -1 0 0 |
| 0 | => (| 0 0 0 1 |, | 0 -1 1 0 |)
| 1 | | 0 0 1 0 |
| 0 1 0 0 |
| 1 0 0 0 |
| 1 | => (| 1 0 0 0 |, | 0 -1 1 0 |)
| 0 | | 0 1 0 0 |
| 0 0 1 0 |
| 0 0 0 1 |
o3 : HashTable
|
i4 : toricChernCharacter E
o4 = HashTable{| -1 0 | => {| 2 |, | -1 |, | -1 |, | 2 |}}
| -1 1 | | -1 | | 1 | | 0 | | -1 |
| 1 -1 | => {| -1 |, | 1 |, | 0 |, | -1 |}
| 0 -1 | | 2 | | -1 | | -1 | | 2 |
| 1 0 | => {| -1 |, | 0 |, | 1 |, | -1 |}
| 0 1 | | -1 | | 1 | | 0 | | -1 |
o4 : HashTable
|
Note that the filtration steps of E are neither ascending nor descending. The method
toricChernCharacter produces nonsense here: $\mathcal{O}_{\mathbb P^2}$ is a direct summand of E, so 0 has to appear in the toric Chern character.
i5 : F = wellformedBundleFiltrations E
o5 = {dimension of the variety => 2 }
number of affine charts => 3
number of rays => 3
rank of the vector bundle => 4
o5 : ToricVectorBundleKlyachko
|
i6 : details F
o6 = HashTable{| -1 | => (| 1 0 1 0 |, | -1 0 0 1 |)}
| -1 | | -1 1 -1 1 |
| 1 0 0 0 |
| -1 1 0 0 |
| 0 | => (| 0 0 1 0 |, | -1 0 0 1 |)
| 1 | | 0 0 0 1 |
| 1 0 0 0 |
| 0 1 0 0 |
| 1 | => (| 0 1 0 0 |, | -1 0 0 1 |)
| 0 | | 1 0 0 0 |
| 0 0 0 1 |
| 0 0 1 0 |
o6 : HashTable
|
i7 : toricChernCharacter F
o7 = HashTable{| -1 0 | => {0, | 0 |, | 0 |, 0}}
| -1 1 | | 1 | | -1 |
| 1 -1 | => {0, | 1 |, | -1 |, 0}
| 0 -1 | | 0 | | 0 |
| 1 0 | => {0, | -1 |, | 1 |, 0}
| 0 1 | | 1 | | -1 |
o7 : HashTable
|
By passing to ascending filtration steps, the result becomes correct.