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 ${\mathbb G}(1,3)$ in two different ways.
We calculate directly:
i1 : G13 = flagBundle({2,2})
o1 = G13
o1 : a flag bundle with subquotient ranks {2:2}
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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 ]
1,1 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
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The above amounts to using the splitting principle.
We also can calculate the total Chern class of the tangent bundle of $G = {\mathbb G}(1,3)$ by realizing $G$ as a smooth quadric in ${\mathbb P}^5$. 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 ${\mathbb P}^5$. Then, we'll substitute $\sigma_1$ into this polynomial, since we know $\sigma_1$ is the hyperplane section.
i4 : P5 = flagBundle({1,5})
o4 = P5
o4 : a flag bundle with subquotient ranks {1, 5}
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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 ]
1,1 2,1 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
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i9 : sigma_1 = chern(1,G13.Bundles#1)
o9 = H
2,1
QQ[][H ..H ]
1,1 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
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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 ]
1,1 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
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