# addFiltration -- changing the filtration matrices of a toric vector bundle in Klyachko's description

## Synopsis

• Usage:
• Inputs:
• L, a list, with matrices over ZZ
• Outputs:

## Description

addFiltration replaces the filtration matrices in E by the matrices in the List L. The matrices in L must be $1$ by $k$ matrices over ZZ, where $k$ is the rank of the vector bundle E. The list has to contain one matrix for each ray of the underlying fan over which E is defined. Note that in E the rays are already sorted and that the filtration matrices in L will be assigned to the rays in that order. To see the order, use rays(ToricVectorBundle).

"The filtration on the vector bundle over a ray is given by the filtration matrix for this ray in the following way: The first index $j$, such that the $i$-th basis vector in the basis over this ray appears in the $j$-th step of the filtration, is the $i$-th entry of the filtration matrix. OR in other words, the $j$-th step step in the filtration is given by all columns of the basis matrix for which the corresponding entry in the filtration matrix is less or equal to $j$."

The matrices need not satisfy the compatibility condition. This can be checked with isVectorBundle.

 i1 : E = toricVectorBundle(2,pp1ProductFan 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 : details E o2 = HashTable{| -1 | => (| 1 0 |, 0)} | 0 | | 0 1 | | 0 | => (| 1 0 |, 0) | -1 | | 0 1 | | 0 | => (| 1 0 |, 0) | 1 | | 0 1 | | 1 | => (| 1 0 |, 0) | 0 | | 0 1 | o2 : HashTable i3 : F = addFiltration(E,{matrix{{1,3}},matrix{{-1,3}},matrix{{2,-3}},matrix{{0,-1}}}) o3 = {dimension of the variety => 2 } number of affine charts => 4 number of rays => 4 rank of the vector bundle => 2 o3 : ToricVectorBundleKlyachko i4 : details F o4 = HashTable{| -1 | => (| 1 0 |, | 2 -3 |)} | 0 | | 0 1 | | 0 | => (| 1 0 |, | 1 3 |) | -1 | | 0 1 | | 0 | => (| 1 0 |, | -1 3 |) | 1 | | 0 1 | | 1 | => (| 1 0 |, | 0 -1 |) | 0 | | 0 1 | o4 : HashTable i5 : isVectorBundle F o5 = true

This means that for example over the first ray the first basis vector of the filtration of F appears at the filtration step 1 and the second at 3.