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chern(ZZ,ZZ,AbstractSheaf) -- Get the Chern class of an abstract sheaf

Synopsis

Description

Chern classes of an abstract sheaf are computed. If called with three arguments as above, a list of the Chern classes cn(A) .. cm(A) are returned. Here 0≤n≤m are integers. chern may also be called without one or both of these integer arguments, in which case just one Chern class, or the total Chern class is returned, respectively. In the following example, we consider two vector bundles A and B of ranks 2 and 3, respectively, on a variety of which we only know that its dimension is 3.
i1 : base(3, Bundle => (A,2,a), Bundle => (B,3,b))

o1 = a variety

o1 : an abstract variety of dimension 3
i2 : chern(B)

o2 = 1 + b  + b  + b
          1    2    3

o2 : QQ[a , a , b , b , b ]
         1   2   1   2   3
i3 : chern(-A)

                2             3
o3 = 1 - a  + (a  - a ) + (- a  + 2a a )
          1     1    2        1     1 2

o3 : QQ[a , a , b , b , b ]
         1   2   1   2   3
i4 : chern(2,A*B)

       2                  2
o4 = 3a  + 3a  + 5a b  + b  + 2b
       1     2     1 1    1     2

o4 : QQ[a , a , b , b , b ]
         1   2   1   2   3
i5 : chern(2,3,B-A)

       2                      3            2
o5 = {a  - a  - a b  + b , - a  + 2a a  + a b  - a b  - a b  + b }
       1    2    1 1    2     1     1 2    1 1    2 1    1 2    3

o5 : List
The next example gives the Chern classes of the twists of a rank-2 vector bundle on the projective plane
i6 : pt=base(n,p,q)

o6 = pt

o6 : an abstract variety of dimension 0
i7 : P2=projectiveSpace(2,pt)

o7 = P2

o7 : a flag bundle with ranks {2, 1}
i8 : E=abstractSheaf(P2,Rank=>2,ChernClass=>1+p*h+q*h^2)

o8 = E

o8 : an abstract sheaf of rank 2 on P2
i9 : chern(E*OO(n*h))

                       2            2
o9 = 1 + (2n + p)h + (n  + n*p + q)h

        QQ[n, p, q][H   , H   , h]
                     1,1   1,2
o9 : -------------------------------
     (H    + h, H    + H   h, H   h)
       1,1       1,2    1,1    1,2

See also