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OldToricVectorBundles :: toricVectorBundle

toricVectorBundle -- the trivial bundle of rank 'k' for a given fan

Synopsis

Description

For a given pure, full dimensional and pointed Fan F the function toricVectorBundle generates the trivial toric vector bundle of rank k.

"If no further options are given then the resulting bundle will be in Klyachko's description: The basis assigned to every ray is the standard basis of $\mathbb{Q}^k$ and the filtration is given by $0$ for all $i<0$ and $\mathbb{Q}^k$ for $i>=0$."

i1 : E = toricVectorBundle(2,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 : details E

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

o2 : HashTable

If the option "Type" => "Kaneyama" is given then the resulting bundle will be in Kaneyama's description: The degree vectors of this bundle are all zero vectors and the transition matrices are all the identity. Note that for Kaneyama's description only complete, pointed fans are implemented and thus a non complete fan will produce an error.

i3 : E = toricVectorBundle(2,pp1ProductFan 2,"Type" => "Kaneyama")

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

o3 : ToricVectorBundleKaneyama
i4 : details E

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

o4 : Sequence

See also

Ways to use toricVectorBundle :

For the programmer

The object toricVectorBundle is a method function with options.