HH^ZZ(ToricVectorBundle,Matrix) -- the i-th cohomology of a toric vector bundle in a given degree

Synopsis

• Function: cohomology

• Usage:

  c = HH_i^E u

• Inputs:
  • i, an integer
  • E, an instance of the type ToricVectorBundle
  • u, a matrix, over ZZ with just one column, giving a weight in the lattice
• Optional inputs:
  • Degree (missing documentation) => ..., default value 0,
• Outputs:
  • c, a module

Description

"Computes the $i$-th cohomology group of the toric vector bundle $E$ of degree $u$ where $u$ must be a one-column matrix giving a point in the lattice of the fan over which $E$ is defined and $i$ must be between $0$ and the dimension of the underlying toric variety."

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

i2 : HH^0 (E,matrix{{1},{0}})

           1
o2 = (QQ[])

o2 : QQ[]-module, free, degrees {{1, 0}}

See also

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