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D.5.2.43 eulerChProj
Procedure from library chern.lib (see chern_lib).
- Usage:
- eulerChProj(n, r, c); n integer, r polynomial (or integer), c list of polynomials
- Return:
- polynomial
- Purpose:
- computes the Euler characteristic of a vector bundle on P_n
in terms of its rank and Chern classes
- Note:
Example:
| | LIB "chern.lib";
ring h=0, (r, c(1..3)), ws(0,1,2,3);
list l=c(1..3);
// the Euler characteristic of a vector bundle on the projective line
print( eulerChProj(1, r, l) );
==> r+c(1)
// the Euler characteristic of a vector bundle on the projective plane
print( eulerChProj(2, r, l) );
==> r+3/2*c(1)+1/2*c(1)^2-c(2)
// the Euler characteristic of a vector bundle on P_3
print( eulerChProj(3, r, l) );
==> r+11/6*c(1)+c(1)^2-2*c(2)+1/6*c(1)^3-1/2*c(1)*c(2)+1/2*c(3)
// assume now that we have a bundle framed at a subplane of P_3
// this implies c(1)=c(2)=0
l= 0, 0, c(3);
// the Euler characteristic is
print( eulerChProj(3, r, l) );
==> r+1/2*c(3)
// which implies that c(3) must be even in this case
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