toricChernCharacter -- computes the toric Chern character of a toric vector bundle

Synopsis

Usage:

c = toricChernCharacter E
Inputs:
- E, a vector bundle on a toric variety (Klyachko's description)
Outputs:
- c, a hash table, whose keys are maximal cones and the value of a key is a list of one column matrices
Consequences:
- The result of toricChernCharacter will be stored as a cacheValue in E.cache#toricChernCharacter.
  As a side product, this method checks whether the sheaf given is locally free this will be stored as a cacheValue in E.cache#isLF. This value is returned by the method isLocallyFree.
  Additionally, toricChernCharacter determines the Cartier index, which will be stored as a cacheValue in E.cache#cartierInd. This value is returned by the method cartierInd.

Description

Given a toric vector bundle in Klyachko's description, toricChernCharacter computes its toric Chern character as introduced in [P].
toricChernCharacter calls internally the method compatibleBases.

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 : 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

Caveat

This method works for a toric reflexive sheaf which is locally Weil (see isLocallyFree for an example if the sheaf is locally Weil but not locally free) on a toric variety, whose fan is covered by simplicial cones of maximal dimension.

Ways to use toricChernCharacter :

toricChernCharacter(ToricVectorBundleKlyachko)

For the programmer

The object toricChernCharacter is a method function with options.