# eulerChi -- the Euler characteristic of a toric vector bundle

## Synopsis

• Usage:
i = eulerChi E
eulerChi(u,E)
• Inputs:
• E, an instance of the type ToricVectorBundle
• u, , with just one column over ZZ representing a degree vector
• Outputs:

## Description

This function computes the Euler characteristic of a vector bundle if only the bundle is given to the function. For this it first computes the set of all degrees that give non-zero cohomology (see deltaE) and then computes the Euler characteristic for each these degrees. If the underlying variety is not complete then this set may not be finite. Thus, for a non-complete toric variety an error is returned.

If in addition a one-column matrix over ZZ, representing a degree vector u, is given, it computes the Euler characteristic of the degree u-part of the vector bundle E. For this the variety need not be complete.

 i1 : E = tangentBundle hirzebruchFan 3 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 : eulerChi(u,E) o3 = 2 i4 : eulerChi E o4 = 6
 i5 : E = tangentBundle(hirzebruchFan 3,"Type" => "Kaneyama") o5 = {dimension of the variety => 2 } number of affine charts => 4 rank of the vector bundle => 2 o5 : ToricVectorBundleKaneyama i6 : u = matrix {{0},{0}} o6 = 0 2 1 o6 : Matrix ZZ <--- ZZ i7 : eulerChi(u,E) o7 = 2 i8 : eulerChi E o8 = 6