# isomorphism -- the isomorphism if the two bundles are isomorphic

## Synopsis

• Usage:
M = isomorphism(E,F)
• Inputs:
• Outputs:
• M, , over the ring over which the two bundles are defined

## Description

Two equivariant vector bundles in Klyachko's description are isomorphic if there exists a simultaneous isomorphism for the filtered vector spaces of all rays. If the two bundles are isomorphic (see areIsomorphic) this function returns the isomorphism. For this, the two bundles must be defined over the same fan.

 i1 : HF = hirzebruchFan 2 o1 = HF o1 : Fan i2 : E = exteriorPower(2, cotangentBundle HF) o2 = {dimension of the variety => 2 } number of affine charts => 4 number of rays => 4 rank of the vector bundle => 1 o2 : ToricVectorBundleKlyachko i3 : F = weilToCartier({-1,-1,-1,-1},HF) o3 = {dimension of the variety => 2 } number of affine charts => 4 number of rays => 4 rank of the vector bundle => 1 o3 : ToricVectorBundleKlyachko i4 : M = isomorphism(E,F) o4 = | 1 | 1 1 o4 : Matrix QQ <--- QQ

• areIsomorphic -- checks if two vector bundles are isomorphic
• base -- the basis matrices for the rays
• filtration -- the filtration matrices of the vector bundle
• details -- the details of a toric vector bundle

## Ways to use isomorphism :

• "isomorphism(ToricVectorBundleKlyachko,ToricVectorBundleKlyachko)"

## For the programmer

The object isomorphism is .