next | previous | forward | backward | up | top | index | toc | Macaulay2 website
OldToricVectorBundles :: eulerChi

eulerChi -- the Euler characteristic of a toric vector bundle

Synopsis

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

See also

Ways to use eulerChi :

For the programmer

The object eulerChi is a method function.