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ToricVectorBundles :: HH^ZZ ToricVectorBundle

HH^ZZ ToricVectorBundle -- the i-th cohomology group of a toric vector bundle

Synopsis

Description

Computes the $i$-th cohomology group of the toric vector bundle $E$. The output is the $i$-th cohomology group as a multigraded module. For this, it computes the set of all degrees that can give non-zero cohomology (see deltaE). This set is finite if the underlying toric variety is complete. If the toric variety is not complete then an error is returned.

The computation of the cohomology groups for a toric vector bundle given in terms of Kaneyama is done by the usual Cech cohomology complex, again separately for every degree $u \in{} M$.

If the option Degree => 1 is used then it displays the number of degrees for which it computes the cohomology. $i$ must be between $0$ and the dimension of the underlying toric variety.

i1 : E = tangentBundle(hirzebruchFan 3,"Type" => "Kaneyama")

o1 = {dimension of the variety => 2 }
      number of affine charts => 4
      rank of the vector bundle => 2

o1 : ToricVectorBundleKaneyama
i2 : HH^0 E

           8
o2 = (QQ[])

o2 : QQ[]-module, free, degrees {{1, 0}, {3, 1}, {2, 1}, 2:{0, 0}, {1, 1}, {-1, 0}, {0, 1}}
i3 : HH^0 (E,Degree => 1)
Number of degrees to calculate: 11
...........
           8
o3 = (QQ[])

o3 : QQ[]-module, free, degrees {{1, 0}, {3, 1}, {2, 1}, 2:{0, 0}, {1, 1}, {-1, 0}, {0, 1}}

"In case the toric vector bundle $E$ is given in Klyachko's description, there is a special exact sequence of finite dimensional vector spaces for every weight $u \in{} M$ whose cohomology groups in degree $i$ are isomorphic to $H^i(X,E)$. This exact sequence can be found in the Klyachko's paper listed on the main page of the documentation."

If the option Degree => 1 is used then it displays the number of degrees for which it computes the cohomology. $i$ must be between $0$ and the dimension of the underlying toric variety.

i4 : E = tangentBundle hirzebruchFan 3

o4 = {dimension of the variety => 2 }
      number of affine charts => 4
      number of rays => 4
      rank of the vector bundle => 2

o4 : ToricVectorBundleKlyachko
i5 : HH^0 E

           8
o5 = (QQ[])

o5 : QQ[]-module, free, degrees {{1, 0}, {3, 1}, {2, 1}, 2:{0, 0}, {1, 1}, {-1, 0}, {0, 1}}
i6 : HH^0 (E,Degree => 1)
Number of degrees to calculate: 11
...........
           8
o6 = (QQ[])

o6 : QQ[]-module, free, degrees {{1, 0}, {3, 1}, {2, 1}, 2:{0, 0}, {1, 1}, {-1, 0}, {0, 1}}

See also