# toricVectorBundle -- the trivial bundle of rank 'k' for a given fan

## Synopsis

• Usage:
E = toricVectorBundle(k,F)
• Inputs:
• k, an integer, strictly positive
• F, an object of class Fan
• Outputs:

## Description

For a given pure, full dimensional and pointed Fan F the function toricVectorBundle generates the trivial toric vector bundle of rank k.

"If no further options are given then the resulting bundle will be in Klyachko's description: The basis assigned to every ray is the standard basis of $\mathbb{Q}^k$ and the filtration is given by $0$ for all $i<0$ and $\mathbb{Q}^k$ for $i>=0$."

 i1 : E = toricVectorBundle(2,projectiveSpaceFan 2) o1 = {dimension of the variety => 2 } number of affine charts => 3 number of rays => 3 rank of the vector bundle => 2 o1 : ToricVectorBundleKlyachko i2 : details E o2 = HashTable{| -1 | => (| 1 0 |, 0)} | -1 | | 0 1 | | 0 | => (| 1 0 |, 0) | 1 | | 0 1 | | 1 | => (| 1 0 |, 0) | 0 | | 0 1 | o2 : HashTable

If the option "Type" => "Kaneyama" is given then the resulting bundle will be in Kaneyama's description: The degree vectors of this bundle are all zero vectors and the transition matrices are all the identity. Note that for Kaneyama's description only complete, pointed fans are implemented and thus a non complete fan will produce an error.

 i3 : E = toricVectorBundle(2,pp1ProductFan 2,"Type" => "Kaneyama") o3 = {dimension of the variety => 2 } number of affine charts => 4 rank of the vector bundle => 2 o3 : ToricVectorBundleKaneyama i4 : details E o4 = (HashTable{0 => (| 1 0 |, 0) }, HashTable{(0, 1) => | 1 0 |}) | 0 1 | | 0 1 | 1 => (| 1 0 |, 0) (0, 2) => | 1 0 | | 0 -1 | | 0 1 | 2 => (| -1 0 |, 0) (1, 3) => | 1 0 | | 0 1 | | 0 1 | 3 => (| -1 0 |, 0) (2, 3) => | 1 0 | | 0 -1 | | 0 1 | o4 : Sequence

• addBaseChange -- changing the transition matrices of a toric vector bundle
• addDegrees -- changing the degrees of a toric vector bundle
• addBase -- changing the basis matrices of a toric vector bundle in Klyachko's description
• addFiltration -- changing the filtration matrices of a toric vector bundle in Klyachko's description
• details -- the details of a toric vector bundle
• regCheck -- checking the regularity condition for a toric vector bundle
• cocycleCheck -- checks if a toric vector bundle fulfills the cocycle condition
• isVectorBundle -- checks if the data does in fact define an equivariant toric vector bundle

## For the programmer

The object toricVectorBundle is .