Description
This function computes the Euler characteristic of a vector bundle if only the bundle is given to the function. For this it first computes the set of all degrees that give non-zero cohomology (see
deltaE) and then computes the Euler characteristic for each these degrees. If the underlying variety is not complete then this set may not be finite. Thus, for a non-complete toric variety an error is returned.
If in addition a one-column matrix over
ZZ, representing a degree vector
u, is given, it computes the Euler characteristic of the degree
u-part of the vector bundle
E. For this the variety need not be complete.
i1 : E = tangentBundle hirzebruchFan 3
o1 = {dimension of the variety => 2 }
number of affine charts => 4
number of rays => 4
rank of the vector bundle => 2
o1 : ToricVectorBundleKlyachko
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i2 : u = matrix {{0},{0}}
o2 = 0
2 1
o2 : Matrix ZZ <-- ZZ
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i3 : eulerChi(u,E)
o3 = 2
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i4 : eulerChi E
o4 = 6
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i5 : E = tangentBundle(hirzebruchFan 3,"Type" => "Kaneyama")
o5 = {dimension of the variety => 2 }
number of affine charts => 4
rank of the vector bundle => 2
o5 : ToricVectorBundleKaneyama
|
i6 : u = matrix {{0},{0}}
o6 = 0
2 1
o6 : Matrix ZZ <-- ZZ
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i7 : eulerChi(u,E)
o7 = 2
|
i8 : eulerChi E
o8 = 6
|