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D.5.2.25 chHom
Procedure from library chern.lib (see chern_lib).
- Usage:
- chHom(r, c, R, C [, N]); r, R polynomials (integers);
c, C lists of polynomials, N integer
- Return:
- list of polynomials
- Purpose:
- computes [up to degree N] the list of Chern classes of the vector bundle Hom(E, F)
in terms of the ranks and the Chern classes of E and F
- Note:
Example:
| | LIB "chern.lib";
ring H = 0, ( r, R, c(1..3), C(1..2) ), dp;
list l=c(1..3);
list L=C(1..2);
// the Chern classes of Hom(E, F) for a vector bundle E of rank 3
// with Chern classes c(1), c(2), c(3)
// and a vector bundle F of rank 2 with Chern classes C(1) and C(2):
print( chHom(3, l, 2, L) );
==> [1]:
==> -2*c(1)+3*C(1)
==> [2]:
==> c(1)^2-5*c(1)*C(1)+3*C(1)^2+2*c(2)+3*C(2)
==> [3]:
==> 2*c(1)^2*C(1)-4*c(1)*C(1)^2+C(1)^3-2*c(1)*c(2)+4*c(2)*C(1)-4*c(1)*C(2)\
+6*C(1)*C(2)-2*c(3)
==> [4]:
==> c(1)^2*C(1)^2-c(1)*C(1)^3-3*c(1)*c(2)*C(1)+3*c(2)*C(1)^2+2*c(1)^2*C(2)\
-6*c(1)*C(1)*C(2)+3*C(1)^2*C(2)+c(2)^2+2*c(1)*c(3)-3*c(3)*C(1)+3*C(2)^2
==> [5]:
==> -c(1)*c(2)*C(1)^2+c(2)*C(1)^3+2*c(1)^2*C(1)*C(2)-2*c(1)*C(1)^2*C(2)+c(\
2)^2*C(1)+2*c(1)*c(3)*C(1)-3*c(3)*C(1)^2-2*c(1)*c(2)*C(2)-2*c(1)*C(2)^2+3\
*C(1)*C(2)^2-2*c(2)*c(3)+6*c(3)*C(2)
==> [6]:
==> c(1)*c(3)*C(1)^2-c(3)*C(1)^3-c(1)*c(2)*C(1)*C(2)+c(2)*C(1)^2*C(2)+c(1)\
^2*C(2)^2-c(1)*C(1)*C(2)^2-c(2)*c(3)*C(1)+c(2)^2*C(2)-2*c(1)*c(3)*C(2)+3*\
c(3)*C(1)*C(2)-2*c(2)*C(2)^2+C(2)^3+c(3)^2
// the first two Chern classes of Hom(E, F) for a vector bundle E of rank r
// with Chern classes c(1) and c(2)
// and a vector bundle G of rank R with Chern classes C(1) and C(2)
// this gives the Chern classes of a tensor product on a complex surface
l=c(1..2);
L=C(1..2);
print( chHom(r, l, R, L, 2 ) );
==> [1]:
==> -R*c(1)+r*C(1)
==> [2]:
==> 1/2*R^2*c(1)^2-r*R*c(1)*C(1)+1/2*r^2*C(1)^2-1/2*R*c(1)^2-1/2*r*C(1)^2+\
R*c(2)+c(1)*C(1)+r*C(2)
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