graphToricChernCharacter -- computes the lines connected the components of the toric Chern character of a toric vector bundle

Synopsis

• Usage:

  g = graphToricChernCharacter E

• Inputs:
  • E, a vector bundle on a toric variety (Klyachko's description)
• Outputs:
  • g, a hash table, whose keys are curve cones and the value of a key is a pair of the connected components
• Consequences:

  • The result of graphToricChernCharacter will be stored as a cacheValue in E.cache#graphToricChernCharacter. It will be used by other methods.

Description

i1 : E = tangentBundle(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 : c = toricChernCharacter E

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

o2 : HashTable

i3 : g = graphToricChernCharacter E

o3 = HashTable{| -1 | => {{| 1  |, | -1 |}, {| 1 |, | 0 |}}}
               | -1 |      | -1 |  | 1  |    | 0 |  | 1 |
               | 0 | => {{| 0  |, | 1  |}, {| -1 |, | 1 |}}
               | 1 |      | -1 |  | -1 |    | 0  |  | 0 |
               | 1 | => {{| 0  |, | 0 |}, {| -1 |, | -1 |}}
               | 0 |      | -1 |  | 1 |    | 0  |  | 1  |

o3 : HashTable

See also

• ToricVectorBundleKlyachko -- the class of all toric vector bundles in Klyachko's description
• groundSet -- computes the ground set of a matroid associated to a toric vector bundle
• compatibleBases -- computes compatible bases for a toric vector bundle
• toricChernCharacter -- computes the toric Chern character of a toric vector bundle