# cotangentBundle -- the cotangent bundle on a toric variety

## Synopsis

• Usage:
E = cotangentBundle F
• Inputs:
• F, an instance of the type Fan
• Optional inputs:
• Type
• Outputs:

## 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