Description
For a given pure, full dimensional and pointed fan
F the function
toricVectorBundle generates the toric vector bundle of rank
k given by the data in the two lists
L1 and
L2.
If no further options are given then the resulting bundle will be in Klyachko's description: The first list
L1 will give the basis matrices and the second list
L2 will give the filtration matrices. Then the resulting vector bundle will have these basis and filtration matrices. The number of matrices in
L1 must match the number of rays of the fan and they must be in GL(
k,$R$) for $R$ being
ZZ or
QQ. They will be assigned to the rays in the order they appear in
rays F. The number of matrices in
L2 must also match the number of rays, and they must be $1$ times
k matrices over
ZZ. The assignment order is the same as for the basis matrices.
Note that the basis and filtration matrices that are given to the function need not satisfy the compatibility condition. This can by checked by using
regCheck.
i1 : L1 = {matrix {{1,0},{0,1}},matrix{{0,1},{1,0}},matrix{{-1,0},{-1,1}}}
o1 = {| 1 0 |, | 0 1 |, | -1 0 |}
| 0 1 | | 1 0 | | -1 1 |
o1 : List
|
i2 : L2 = {matrix {{-1,0}},matrix{{-2,-1}},matrix{{0,1}}}
o2 = {| -1 0 |, | -2 -1 |, | 0 1 |}
o2 : List
|
i3 : E = toricVectorBundle(2,projectiveSpaceFan 2,L1,L2)
o3 = {dimension of the variety => 2 }
number of affine charts => 3
number of rays => 3
rank of the vector bundle => 2
o3 : ToricVectorBundleKlyachko
|
i4 : details E
o4 = HashTable{| -1 | => (| 1 0 |, | -1 0 |)}
| -1 | | 0 1 |
| 0 | => (| 0 1 |, | -2 -1 |)
| 1 | | 1 0 |
| 1 | => (| -1 0 |, | 0 1 |)
| 0 | | -1 1 |
o4 : HashTable
|
If the option
"Type" => "Kaneyama" is given then the resulting bundle will be in Kaneyama's description; Note that this is only implemented for complete, pointed fans: The first list
L1 will give the degree matrices and the second list
L2 will give the transition matrices. The number of matrices in
L1 must match the number of maximal cones of the fan and they must be $n$ times
k matrices over
ZZ. They will be assigned to the cones in the order they appear in
maxCones F. The number of matrices in
L2 must match the number of pairs of maximal cones that intersect in a common codimension-one face and must all be in GL(
k,
QQ). They will be assigned to the pairs $(i,j)$ in lexicographic order.
Note that the degrees and transition matrices that are given to the function need not satisfy the regularity or the cocycle condition. These can be checked by using
regCheck and
cocycleCheck.
i5 : L1 = {matrix {{1,0},{0,1}},matrix{{0,1},{1,0}},matrix{{-1,0},{-1,1}}}
o5 = {| 1 0 |, | 0 1 |, | -1 0 |}
| 0 1 | | 1 0 | | -1 1 |
o5 : List
|
i6 : L2 = {matrix {{-1,0},{0,-1}},matrix{{0,1},{1,0}},matrix{{0,-1},{-1,0}}}
o6 = {| -1 0 |, | 0 1 |, | 0 -1 |}
| 0 -1 | | 1 0 | | -1 0 |
o6 : List
|
i7 : E = toricVectorBundle(2,projectiveSpaceFan 2,L1,L2,"Type" => "Kaneyama")
o7 = {dimension of the variety => 2 }
number of affine charts => 3
rank of the vector bundle => 2
o7 : ToricVectorBundleKaneyama
|
i8 : details E
o8 = (HashTable{0 => (| -1 0 |, | 1 0 |) }, HashTable{(0, 1) => | -1 0 |})
| -1 1 | | 0 1 | | 0 -1 |
1 => (| 1 0 |, | 0 1 |) (0, 2) => | 0 1 |
| 0 1 | | 1 0 | | 1 0 |
2 => (| 1 -1 |, | -1 0 |) (1, 2) => | 0 -1 |
| 0 -1 | | -1 1 | | -1 0 |
o8 : Sequence
|