filtration -- the filtration matrices of the vector bundle

Synopsis

• Usage:

  f = filtration E

• Inputs:
  • E, a vector bundle on a toric variety (Klyachko's description)
• Outputs:
  • f, a hash table

Description

For each ray of the fan there is a filtration matrix. If the bundle has rank $k$ then this is a one row matrix over ZZ with $k$ entries. This defines the filtration on the corresponding base matrix (see base) such that the $j$-th filtration is generated by all columns of the base matrix for which the entry in the same column of the filtration matrix is less or equal to $j$.

i1 : E = tangentBundle hirzebruchFan 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 : filtration E

o2 = HashTable{| -1 | => | -1 0 |}
               | 2  |
               | 0  | => | -1 0 |
               | -1 |
               | 0 | => | -1 0 |
               | 1 |
               | 1 | => | -1 0 |
               | 0 |

o2 : HashTable

So in this example for each ray the first column of the basis appears at -1 and the second at 0.

See also

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