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ToricVectorBundles :: isVectorBundle

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

Synopsis

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.

E = toricVectorBundle(2,pp1ProductFan 2)
E = addBase(E,{matrix{{1,2},{3,1}},matrix{{-1,0},{3,1}},matrix{{1,2},{-3,-1}},matrix{{-1,0},{-3,-1}}})
isVectorBundle E
F = toricVectorBundle(1,normalFan crossPolytope 3)
F = addFiltration(F,apply({2,1,1,2,2,1,1,2}, i -> matrix {{i}}))
isVectorBundle F

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).

E = toricVectorBundle(2,pp1ProductFan 2,"Type" => "Kaneyama")
isVectorBundle E
E = addBaseChange(E,{matrix{{1,2},{3,1}},matrix{{-1,0},{3,1}},matrix{{1,2},{-3,-1}},matrix{{-1,0},{-3,-1}}})
isVectorBundle E

See also

Ways to use isVectorBundle :