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

Caveat

isVectorBundle is known to produce incorrect output for Klyachko bundles. The user is recommended to instead use isLocallyFree from the package PositivityToricBundles.

See also

Ways to use isVectorBundle:

  • isVectorBundle(ToricVectorBundle)

For the programmer

The object isVectorBundle is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/ToricVectorBundles.m2:3231:0.