addFiltration -- changing the filtration matrices of a toric vector bundle in Klyachko's description

Synopsis

• Usage:

  F = addFiltration(E,L)

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

Description

addFiltration replaces the filtration matrices in E by the matrices in the List L. The matrices in L must be $1$ by $k$ matrices over ZZ, where $k$ is the rank of the vector bundle E. 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 filtration matrices in L will be assigned to the rays in that order. To see the order, use rays(ToricVectorBundle).

"The filtration on the vector bundle over a ray is given by the filtration matrix for this ray in the following way: The first index $j$, such that the $i$-th basis vector in the basis over this ray appears in the $j$-th step of the filtration, is the $i$-th entry of the filtration matrix. OR in other words, the $j$-th step step in the filtration is given by all columns of the basis matrix for which the corresponding entry in the filtration matrix is less or equal to $j$."

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 = addFiltration(E,{matrix{{1,3}},matrix{{-1,3}},matrix{{2,-3}},matrix{{0,-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 0 |, | 2 -3 |)}
               | 0  |     | 0 1 |
               | 0  | => (| 1 0 |, | 1 3 |)
               | -1 |     | 0 1 |
               | 0 | => (| 1 0 |, | -1 3 |)
               | 1 |     | 0 1 |
               | 1 | => (| 1 0 |, | 0 -1 |)
               | 0 |     | 0 1 |

o4 : HashTable

i5 : isVectorBundle F

o5 = true

This means that for example over the first ray the first basis vector of the filtration of F appears at the filtration step 1 and the second at 3.

See also

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