areIsomorphic -- checks if two vector bundles are isomorphic
Synopsis
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- Usage:
b = areIsomorphic(E,F)
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Inputs:
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Outputs:
Description
E and
F must be vector bundles over the same fan and the filtrations must be defined over the same ring. 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 = HF
o1 : Fan
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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
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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
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i4 : areIsomorphic(E,F)
o4 = true
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To obtain the isomorphism, if two bundles are isomorphic use
isomorphism.
Caveat
If E and F are defined over different rings (e.g. QQ and ZZ) then areIsomorphic(E,F) will return false. Likewise, if the bundles are only defined over ZZ, the function will check for an isomorphism of the filtrations over ZZ.
See also
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isomorphism -- the isomorphism if the two bundles are isomorphic
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base -- the basis matrices for the rays
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filtration -- the filtration matrices of the vector bundle
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details -- the details of a toric vector bundle
Ways to use areIsomorphic :
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areIsomorphic(ToricVectorBundleKlyachko,ToricVectorBundleKlyachko)