If the fan F is pure, of full dimension and smooth, then the function generates the tangent 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 = pp1ProductFan 2
o1 = F
o1 : Fan
i2 : E = tangentBundle F
o2 = {dimension of the variety => 2 }
number of affine charts => 4
number of rays => 4
rank of the vector bundle => 2
o2 : ToricVectorBundleKlyachko
If the option "Type" => "Kaneyama" is given then the resulting bundle will be in Kaneyama's description:
i4 : F = pp1ProductFan 2
o4 = F
o4 : Fan
i5 : E = tangentBundle(F,"Type" => "Kaneyama")
o5 = {dimension of the variety => 2 }
number of affine charts => 4
rank of the vector bundle => 2
o5 : ToricVectorBundleKaneyama