Macaulay2 » Documentation
Packages » OldToricVectorBundles :: HH^ZZ ToricVectorBundle
next | previous | forward | backward | up | index | toc

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

- Function: cohomology
- Usage:
    `c = HH^i E`
- Inputs:
  - i, an integer
  - T, an instance of the type ToricVectorBundle
- Optional inputs:
  - Degree => ..., default value 0,
- Outputs:
  - c, a module

## 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}, 2:{0, 0}, {1, 0}, {0, 1}, {1, 1}, {2, 1}, {3, 1}}

    i3 : HH^0 (E,Degree => 1)
    Number of degrees to calculate: 11
    ...........
               8
    o3 = (QQ[])

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

    i6 : HH^0 (E,Degree => 1)
    Number of degrees to calculate: 11
    ...........
               8
    o6 = (QQ[])

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

## See also

- ring(ToricVectorBundle) -- the graded ring of the bundle
- deltaE -- the polytope of possible degrees that give non zero cohomology
- HH^ZZ(ToricVectorBundle,Matrix) -- the i-th cohomology of a toric vector bundle in a given degree
- HH^ZZ(ToricVectorBundle,List) -- the i-th cohomology of a toric vector bundle for a given list of degrees
- hh^ZZ ToricVectorBundle
- eulerChi -- the Euler characteristic of a toric vector bundle

## Ways to use this method:

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

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/OldToricVectorBundles.m2:2537:0.