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Book3264Examples > Intersection Theory Section 5.4.4

Intersection Theory Section 5.4.4 -- Bundles on Grassmannians

We already know everything necessary to calculate chern classes of bundles on Grassmannians.

As an example, we can do:

Exercise 5.17: Calculate the chern classes of the tangent bundle to G(1,3) in two different ways.

We calculate directly:

i1 : G13 = flagBundle({2,2})

o1 = G13

o1 : a flag bundle with subquotient ranks {2:2}
i2 : T = tangentBundle(G13)

o2 = T

o2 : an abstract sheaf of rank 4 on G13
i3 : chern T

                   2                    2
o3 = 1 + 4H    + 7H    + 12H   H    + 6H
           2,1     2,1      2,1 2,2     2,2

                             QQ[][H   , H   , H   , H   ]
                                   1,1   1,2   2,1   2,2
o3 : ---------------------------------------------------------------------------
     (- H    - H   , - H    - H   H    - H   , - H   H    - H   H   , -H   H   )
         1,1    2,1     1,2    1,1 2,1    2,2     1,2 2,1    1,1 2,2    1,2 2,2

The above amounts to using the splitting principle.

We also can calculate the total Chern class of the tangent bundle of G = G(1,3) by realizing G as a smooth quadric in 5. The plan is the following: first, we’ll calculate the total Chern class of the tangent bundle in terms of powers of the hyperplane section of G in 5. Then, we’ll substitute σ1 into this polynomial, since we know σ1 is the hyperplane section.

i4 : P5 = flagBundle({1,5})

o4 = P5

o4 : a flag bundle with subquotient ranks {1, 5}
i5 : TP5 = tangentBundle(P5)

o5 = TP5

o5 : an abstract sheaf of rank 5 on P5
i6 : O1 = dual(P5.Bundles#0)

o6 = O1

o6 : an abstract sheaf of rank 1 on P5
i7 : O2 = O1^**2

o7 = O2

o7 : an abstract sheaf of rank 1 on P5
i8 : TG = chern(TP5 - O2) -- total Chern class of TG in terms of the hyperplane section

o8 = 1 + 4H    + 7H    + 6H    + 3H
           2,1     2,2     2,3     2,4

                                    QQ[][H   , H   , H   , H   , H   , H   ]
                                          1,1   2,1   2,2   2,3   2,4   2,5
o8 : ------------------------------------------------------------------------------------------------------
     (- H    - H   , - H   H    - H   , - H   H    - H   , - H   H    - H   , - H   H    - H   , -H   H   )
         1,1    2,1     1,1 2,1    2,2     1,1 2,2    2,3     1,1 2,3    2,4     1,1 2,4    2,5    1,1 2,5
i9 : sigma_1 = chern(1,G13.Bundles#1)

o9 = H
      2,1

                             QQ[][H   , H   , H   , H   ]
                                   1,1   1,2   2,1   2,2
o9 : ---------------------------------------------------------------------------
     (- H    - H   , - H    - H   H    - H   , - H   H    - H   H   , -H   H   )
         1,1    2,1     1,2    1,1 2,1    2,2     1,2 2,1    1,1 2,2    1,2 2,2
i10 : 1 + sum(1..4, i -> coefficient(chern(i,P5.Bundles#1),TG) * ((sigma_1)^i))

                    2                    2
o10 = 1 + 4H    + 7H    + 12H   H    + 6H
            2,1     2,1      2,1 2,2     2,2

                              QQ[][H   , H   , H   , H   ]
                                    1,1   1,2   2,1   2,2
o10 : ---------------------------------------------------------------------------
      (- H    - H   , - H    - H   H    - H   , - H   H    - H   H   , -H   H   )
          1,1    2,1     1,2    1,1 2,1    2,2     1,2 2,1    1,1 2,2    1,2 2,2