1
vote
0answers
89 views

Grassmannian bundle: any good reference?

I have met the notion of Grassmannian bundle of a vector bundle over a variety in intersection theory, but anywhere I look I just find a brief recall of how the stalks look like (my references so far ...
1
vote
1answer
77 views

Canonical line bundle over a projective bundle

The following is an excerpt from the Atiyah's K-Theory. If $E$ is a vector bundle over $X$ then each point $a\in P(E)_{x}=P(E_{x})$ represents a one-dimensional subspace $H_{x}^{*}\subset E_{x}$. ...
1
vote
0answers
36 views

Non-(stable)-triviality of the tautological bundles

The tautological vector bundle $\gamma_k(\mathbb{K}^N)$ over the Grassmann manifold $G_k(\mathbb{K}^N)$ of all $k$-planes in $\mathbb{K}^N$ (for $\mathbb{K} = \mathbb{R}$, $\mathbb{C}$ or ...
5
votes
1answer
350 views

Tautological vector bundle over $G_1(\mathbb{R^2})$ isomorphic to the Möbius bundle

Let $V$ be a finite dimensional vector space, and let $G_k(V)$ be the Grassmannian of $k$-dimensional subspaces of $V$. Let $T$ be the disjoint union of all these $k$-dimensional subspaces and ...