deltaE -- the polytope of possible degrees that give non zero cohomology

Synopsis

• Usage:

  P = deltaE E

• Inputs:
  • E, an instance of the type ToricVectorBundle
• Outputs:
  • P, a convex polyhedron

Description

For a toric vector bundle over a complete toric variety there is a finite set of degrees $u$ such that the degree $u$ part of the cohomology of the vector bundle is non-zero. This function computes a polytope $\Delta_E$, such that these degrees are contained in this polytope. If the underlying toric variety is not complete then an error is returned.

i1 : E = toricVectorBundle(2,pp1ProductFan 2, "Type" => "Kaneyama")

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

o1 : ToricVectorBundleKaneyama

i2 : P = deltaE E

o2 = P

o2 : Polyhedron

i3 : vertices P

o3 = 0

              2        1
o3 : Matrix QQ  <--- QQ

i4 : E1 = tangentBundle projectiveSpaceFan 2

o4 = {dimension of the variety => 2 }
      number of affine charts => 3
      number of rays => 3
      rank of the vector bundle => 2

o4 : ToricVectorBundleKlyachko

i5 : P1 = deltaE E1

o5 = P1

o5 : Polyhedron

i6 : vertices P1

o6 = | -1 1 0  1  0 -1 |
     | 0  0 -1 -1 1 1  |

              2        6
o6 : Matrix QQ  <--- QQ

See also

• eulerChi -- the Euler characteristic of a toric vector bundle
• HH^ZZ ToricVectorBundle -- the i-th cohomology group of a toric vector bundle
• hh^ZZ ToricVectorBundle -- the rank of the i-th cohomology group of a toric vector bundle