3
votes
2answers
131 views

Why is $\mathbb{P}(T\mathbb{P}^2)$ isomorphic to the space of complete flags $GL_3/B$?

In one the the answers to this thread " Can one embedd the projectivezed tangent space of CP^2 in a projective space? " it was mentioned that " $\mathbb{P}(T\mathbb{P}^2)$ isomorphic to the variety ...
1
vote
1answer
87 views

Non-equivariant vector bundles over complex projective $N$-space

From Grothendieck's lemma, we know that all holomorphic vector bundles over the complex projective line are direct sums of line bundles, and so, are $SU(2)$-equivariant. I wonder, do there exist ...
1
vote
1answer
308 views

Computing relative Lie algebra cohomology (as appears in Borel-Weil-Bott theorem)

Suppose $G$ is a complex Lie group, $P$ a Borel subgroup, $E$ a representation of $P$ that induces a vector bundle ${\cal E}$ over $G/P$. The general version of Borel-Weil-Bott theorem, as stated in ...
2
votes
2answers
170 views

Are all Equivariant Bundles of a Total Flag Manifold Constructable from Line Bundles?

As we all know, for any homogeneous space $G/H$ we have that the equivariant vector bundles over $G/H$ are characterized by the representations of $H$. Thus, for the the complex projective line $CP^1 ...