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# cotangentBundle -- the cotangent bundle on a toric variety

- Usage:
    `E = cotangentBundle F`
- Inputs:
  - F, an instance of the type Fan
- Optional inputs:
  - Type
- Outputs:
  - E, a vector bundle on a toric variety (Kaneyama's description) or a vector bundle on a toric variety (Klyachko's description)

## Description

If the fan F is pure, of full dimension and smooth, then the function generates the cotangent bundle of the toric variety given by F. If no further options are given then the resulting bundle will be in Klyachko's description:

    i1 : F = projectiveSpaceFan 2

    o1 = {ambient dimension => 2         }
          number of generating cones => 3
          number of rays => 3
          top dimension of the cones => 2

    o1 : Fan

    i2 : E = tangentBundle F

    o2 = {dimension of the variety => 2 }
          number of affine charts => 3
          number of rays => 3
          rank of the vector bundle => 2

    o2 : ToricVectorBundleKlyachko

    i3 : details E

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

    o3 : HashTable

If the option "Type" => "Kaneyama" is given then the resulting bundle will be in Kaneyama's description:

    i4 : F = projectiveSpaceFan 2

    o4 = {ambient dimension => 2         }
          number of generating cones => 3
          number of rays => 3
          top dimension of the cones => 2

    o4 : Fan

    i5 : E = tangentBundle(F,"Type" => "Kaneyama")

    o5 = {dimension of the variety => 2 }
          number of affine charts => 3
          rank of the vector bundle => 2

    o5 : ToricVectorBundleKaneyama

    i6 : details E

    o6 = (HashTable{0 => (| -1 0 |, | 1 1  |)}, HashTable{(0, 1) => | -1 0 |})
                          | -1 1 |  | 0 -1 |                        | -1 1 |
                    1 => (| 1 0 |, | -1 0  |)             (0, 2) => | 1 -1 |
                          | 0 1 |  | 0  -1 |                        | 0 -1 |
                    2 => (| 1 -1 |, | 0 -1 |)             (1, 2) => | 0 -1 |
                          | 0 -1 |  | 1 1  |                        | 1 -1 |

    o6 : Sequence

## See also

- tangentBundle -- the tangent bundle on a toric variety

## Ways to use cotangentBundle:

- `cotangentBundle(Fan)`

## For the programmer

The object cotangentBundle is a method function with options.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/OldToricVectorBundles.m2:2667:0.