Macaulay2 » Documentation
Packages » ToricVectorBundles :: cotangentBundle
next | previous | forward | backward | up | index | toc

cotangentBundle -- the cotangent bundle on a toric variety

Synopsis

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 = F

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 = F

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

Ways to use cotangentBundle :

For the programmer

The object cotangentBundle is a method function with options.