addBase -- changing the basis matrices of a toric vector bundle in Klyachko's description

Synopsis

• Usage:

  F = addBase(E,L)

• Inputs:
  • E, a vector bundle on a toric variety (Klyachko's description)
  • L, a list, with matrices over ZZ or QQ
• Outputs:
  • F, a vector bundle on a toric variety (Klyachko's description)

Description

addBase replaces the basis matrices in E by the matrices in the List L. The matrices in L must be in GL($k,R$), where $k$ is the rank of the vector bundle E and $R$ is ZZ or QQ. The list has to contain one matrix for each ray of the underlying fan over which E is defined. Note that in E the rays are already sorted and that the basis matrices in L will be assigned to the rays in that order. To see the order use rays(ToricVectorBundle).

The matrices need not satisfy the compatibility condition. This can be checked with isVectorBundle.

i1 : E = toricVectorBundle(2,pp1ProductFan 2)

o1 = {dimension of the variety => 2 }
      number of affine charts => 4
      number of rays => 4
      rank of the vector bundle => 2

o1 : ToricVectorBundleKlyachko

i2 : details E

o2 = HashTable{| -1 | => (| 1 0 |, 0)}
               | 0  |     | 0 1 |
               | 0  | => (| 1 0 |, 0)
               | -1 |     | 0 1 |
               | 0 | => (| 1 0 |, 0)
               | 1 |     | 0 1 |
               | 1 | => (| 1 0 |, 0)
               | 0 |     | 0 1 |

o2 : HashTable

i3 : F = addBase(E,{matrix{{1,2},{3,1}},matrix{{-1,0},{3,1}},matrix{{1,2},{-3,-1}},matrix{{-1,0},{-3,-1}}})

o3 = {dimension of the variety => 2 }
      number of affine charts => 4
      number of rays => 4
      rank of the vector bundle => 2

o3 : ToricVectorBundleKlyachko

i4 : details F

o4 = HashTable{| -1 | => (| 1  2  |, 0)}
               | 0  |     | -3 -1 |
               | 0  | => (| 1 2 |, 0)
               | -1 |     | 3 1 |
               | 0 | => (| -1 0 |, 0)
               | 1 |     | 3  1 |
               | 1 | => (| -1 0  |, 0)
               | 0 |     | -3 -1 |

o4 : HashTable

i5 : isVectorBundle F

o5 = true

See also

• base -- the basis matrices for the rays
• addFiltration -- changing the filtration matrices of a toric vector bundle in Klyachko's description
• isVectorBundle -- checks if the data does in fact define an equivariant toric vector bundle