# 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 ${\mathbb G}(1,3)$ in two different ways.

We calculate directly:

 i1 : G13 = flagBundle({2,2}) o1 = G13 o1 : a flag bundle with subquotient ranks {2:2} 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

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} 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 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 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