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

## Synopsis

• Scripted functor: hh
• Usage:
d = hh^i E
d = hh^i (E,u)
• Inputs:
• i, an integer
• E, an instance of the type ToricVectorBundle, and an optional u, over ZZ, giving a point in the lattice of the fan
• Outputs:

## Description

hh^i computes the rank of the $i$-th cohomology group. If no further argument is given then it returns the rank of the complete cohomology group. 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.

If in addition a one column matrix $u$ over ZZ is given it returns the rank of the degree $u$ part of the cohomology group. For this the variety need not be complete.

 i1 : E = tangentBundle hirzebruchFan 2 o1 = {dimension of the variety => 2 } number of affine charts => 4 number of rays => 4 rank of the vector bundle => 2 o1 : ToricVectorBundleKlyachko i2 : u = matrix{{0},{0}} o2 = 0 2 1 o2 : Matrix ZZ <--- ZZ i3 : hh^0 (E,u) o3 = 2 i4 : hh^0 E o4 = 7