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Book3264Examples > Intersection Theory Section 5.3

Intersection Theory Section 5.3 -- Operations on vector bundles

Schubert2 has all of the basic operations on vector bundles and their Chern classes built in. A full list of all of the available operations can be found in the documentation for AbstractSheaf. A few examples:

Direct Sums:

i1 : GG24 = flagBundle({3,2})

o1 = GG24

o1 : a flag bundle with subquotient ranks {3, 2}
i2 : (S,Q) = GG24.Bundles

o2 = (S, Q)

o2 : Sequence
i3 : B1 = S + Q --direct sum of S+Q

o3 = B1

o3 : an abstract sheaf of rank 5 on GG24
i4 : chern B1

o4 = 1

                                            QQ[][H   ..H   , H   ..H   ]
                                                  1,1   1,3   2,1   2,2
o4 : ---------------------------------------------------------------------------------------------------------
     (- H    - H   , - H    - H   H    - H   , - H    - H   H    - H   H   , - H   H    - H   H   , -H   H   )
         1,1    2,1     1,2    1,1 2,1    2,2     1,3    1,2 2,1    1,1 2,2     1,3 2,1    1,2 2,2    1,3 2,2

Note that the Chern class of $S+Q$ is the same as that of the trivial bundle, since $S$ and $Q$ fit into an exact sequence whose middle term is trivial (see Prop 5.5).

Tensor Products:

i5 : B2 = S ** Q --tensor product of S and Q

o5 = B2

o5 : an abstract sheaf of rank 6 on GG24
i6 : chern B2

                  2                 3                      2           2     
o6 = 1 + H    + (H    + H   ) + (- H    + 4H   H   ) + (- H   H    + 4H   ) -
          2,1     2,1    2,2        2,1     2,1 2,2        2,1 2,2     2,2   
     ------------------------------------------------------------------------
           2       3
     2H   H    + 2H
       2,1 2,2     2,2

                                            QQ[][H   ..H   , H   ..H   ]
                                                  1,1   1,3   2,1   2,2
o6 : ---------------------------------------------------------------------------------------------------------
     (- H    - H   , - H    - H   H    - H   , - H    - H   H    - H   H   , - H   H    - H   H   , -H   H   )
         1,1    2,1     1,2    1,1 2,1    2,2     1,3    1,2 2,1    1,1 2,2     1,3 2,1    1,2 2,2    1,3 2,2

Duals:

i7 : B3 = dual(S) ** Q

o7 = B3

o7 : an abstract sheaf of rank 6 on GG24
i8 : chern B3

                     2                3         2            2     
o8 = 1 + 5H    + (11H    + H   ) + 15H    + (35H   H    - 10H   ) +
           2,1       2,1    2,2       2,1       2,1 2,2      2,2   
     ------------------------------------------------------------------------
            2        3
     30H   H    + 10H
        2,1 2,2      2,2

                                            QQ[][H   ..H   , H   ..H   ]
                                                  1,1   1,3   2,1   2,2
o8 : ---------------------------------------------------------------------------------------------------------
     (- H    - H   , - H    - H   H    - H   , - H    - H   H    - H   H   , - H   H    - H   H   , -H   H   )
         1,1    2,1     1,2    1,1 2,1    2,2     1,3    1,2 2,1    1,1 2,2     1,3 2,1    1,2 2,2    1,3 2,2

Note that B3 is the tangent bundle to $\mathbb{G}(2,4)$.

Pullbacks:

Currently Schubert2 has few morphisms implemented, but given a morphism of abstract varieties, it is easy to pull back vector bundles:

i9 : GG13 = flagBundle({2,2})

o9 = GG13

o9 : a flag bundle with subquotient ranks {2:2}
i10 : f = GG13 / point -- The structure map of G13

o10 = f

o10 : a map to point from GG13
i11 : B = abstractSheaf(point,Rank=>2) -- The trivial vector bundle of rank 2 over point

o11 = B

o11 : an abstract sheaf of rank 2 on point
i12 : f^* B --Pulls back to a trivial bundle of rank 2 on G13

o12 = a sheaf

o12 : an abstract sheaf of rank 2 on GG13