integral -- compute an integral (push forward a cycle to the base)

Synopsis

Usage:
integral f
Inputs:
- f, a ring element, a cycle class in the intersection ring of an abstract variety
Outputs:
- a ring element, the integral of f.

Description

In this example we compute the number of lines meeting four lines in space.

i1 : G = flagBundle {2,2}

o1 = G

o1 : a flag bundle with ranks {2, 2}

i2 : dim G

o2 = 4

i3 : A = intersectionRing G

o3 = A

o3 : QuotientRing

i4 : f = (chern_1 OO_G(1))^4

       2
o4 = 2H
       2,2

o4 : A

i5 : integral f

o5 = 2

Normally this integral will be an element of the intersection ring of point, and thus will essentially be a rational number, but it could be in any base variety, and would be obtained by pushing forward along the structure maps of flag bundles until the (ultimate) base variety is reached.

i6 : pt = base n

o6 = pt

o6 : an abstract variety of dimension 0

i7 : F = flagBundle_{2,2} pt

o7 = F

o7 : a flag bundle with ranks {2, 2}

i8 : integral (chern_1 OO_F(n))^4

       4
o8 = 2n

o8 : QQ[n]