areIsomorphic -- checks if two vector bundles are isomorphic

Synopsis

• Usage:

  b = areIsomorphic(E,F)

• Inputs:
  • E, a vector bundle on a toric variety (Klyachko's description)
  • F, a vector bundle on a toric variety (Klyachko's description)
• Outputs:
  • b, a Boolean value, whether E and F are isomorphic

Description

E and F must be vector bundles over the same fan. Two equivariant vector bundles in Klyachko's description are isomorphic if there exists a simultaneous isomorphism for the filtered vector spaces of all rays. The method then returns whether the bundles are isomorphic.

i1 : HF = hirzebruchFan 2

o1 = {ambient dimension => 2         }
      number of generating cones => 4
      number of rays => 4
      top dimension of the cones => 2

o1 : Fan

i2 : E = exteriorPower(2, cotangentBundle HF)

o2 = {dimension of the variety => 2 }
      number of affine charts => 4
      number of rays => 4
      rank of the vector bundle => 1

o2 : ToricVectorBundleKlyachko

i3 : F = weilToCartier({-1,-1,-1,-1},HF)

o3 = {dimension of the variety => 2 }
      number of affine charts => 4
      number of rays => 4
      rank of the vector bundle => 1

o3 : ToricVectorBundleKlyachko

i4 : areIsomorphic(E,F)

o4 = true

To obtain the isomorphism, if two bundles are isomorphic use isomorphism.

See also

• isomorphism -- the isomorphism if the two bundles are isomorphic
• base -- the basis matrices for the rays
• filtration -- the filtration matrices of the vector bundle
• details -- the details of a toric vector bundle