isVectorBundle -- checks if the data does in fact define an equivariant toric vector bundle

Synopsis

• Usage:

  b = isVectorBundle E

• Inputs:
  • E, an instance of the type ToricVectorBundle
• Outputs:
  • b, a Boolean value, whether E defines a toric vector bundle

Description

If E is in Klyachko's description then the data in E defines an equivariant toric vector on the toric variety if and only if for each maximal cone exists a decomposition into torus eigenspaces of the bundle. See Sam Payne's Moduli of toric vector bundles, Compositio Math. 144, 2008. Section 2.3. This uses the two functions findWeights and existsDecomposition.

i1 : E = toricVectorBundle(2,pp1ProductFan 2)

o1 = {dimension of the variety => 2 }
      number of affine charts => 4
      number of rays => 4
      rank of the vector bundle => 2

o1 : ToricVectorBundleKlyachko

i2 : E = addBase(E,{matrix{{1,2},{3,1}},matrix{{-1,0},{3,1}},matrix{{1,2},{-3,-1}},matrix{{-1,0},{-3,-1}}})

o2 = {dimension of the variety => 2 }
      number of affine charts => 4
      number of rays => 4
      rank of the vector bundle => 2

o2 : ToricVectorBundleKlyachko

i3 : isVectorBundle E

o3 = true

i4 : F = toricVectorBundle(1,normalFan crossPolytope 3)

o4 = {dimension of the variety => 3 }
      number of affine charts => 6
      number of rays => 8
      rank of the vector bundle => 1

o4 : ToricVectorBundleKlyachko

i5 : F = addFiltration(F,apply({2,1,1,2,2,1,1,2}, i -> matrix {{i}}))

o5 = {dimension of the variety => 3 }
      number of affine charts => 6
      number of rays => 8
      rank of the vector bundle => 1

o5 : ToricVectorBundleKlyachko

i6 : isVectorBundle F

o6 = false

If E is in Kaneyama's description then data in E defines an equivariant toric vector bundle on the toric variety if and only if it satisfies the regularity and the cocycle condition (See cocycleCheck and regCheck).

i7 : E = toricVectorBundle(2,pp1ProductFan 2,"Type" => "Kaneyama")

o7 = {dimension of the variety => 2 }
      number of affine charts => 4
      rank of the vector bundle => 2

o7 : ToricVectorBundleKaneyama

i8 : isVectorBundle E

o8 = true

i9 : E = addBaseChange(E,{matrix{{1,2},{3,1}},matrix{{-1,0},{3,1}},matrix{{1,2},{-3,-1}},matrix{{-1,0},{-3,-1}}})

o9 = {dimension of the variety => 2 }
      number of affine charts => 4
      rank of the vector bundle => 2

o9 : ToricVectorBundleKaneyama

i10 : isVectorBundle E

o10 = false

See also

• findWeights -- finds the possible weight vectors for the maximal cones
• existsDecomposition -- checks if a list of matrices of weight vectors for each maximal cone admits a decomposition
• addBase -- changing the basis matrices of a toric vector bundle in Klyachko's description
• addFiltration -- changing the filtration matrices of a toric vector bundle in Klyachko's description
• cocycleCheck -- checks if a toric vector bundle fulfills the cocycle condition
• regCheck -- checking the regularity condition for a toric vector bundle
• addBaseChange -- changing the transition matrices of a toric vector bundle
• addDegrees -- changing the degrees of a toric vector bundle
• details -- the details of a toric vector bundle