This is exercise 2.1.1. in [AF23].
Example
For \(d\leq n\), let \(p:E\to Y\) be a rank \(n\) vector bundle over \(Y\) and \(\operatorname{Fr}(d, E)\to Y\) be the frame bundle of \(E\), i.e. the bundle whose fiber over \(y\in Y\) is \[\big\{(v_1,...,v_d)~ |~ v_1,...,v_d \text{ are linearly independen in the fiber } E_y\big\}.\] There is a right \(\GL_d\)-action on this space. We can form the Grassmann bundle of \(E\), \(\Gr(d, E)\), which comes with a tautological bundle \(S \subset p^*E\) over it. Show that \(\operatorname{Fr}(d,E)\times^{\GL_d}\mathbb C^d\) is naturally identified with \(S\).
Proof
For clarity here are definitions.
- \(\Gr(d, E) = \{(y,V) ~|~ y\in Y, V\subset E_y \text{ is a $d$-dimensional subspace}\}\). It has a projection \(\pi:(y,V)\mapsto y\) down to \(Y\).
- \(p^*E = \{(y,V,u) \in \Gr(d, E)\times E ~|~ \pi(y,V) = p(u) \iff u\in E_y\}\), this is a bundle over \(\Gr(d,E)\).
- \(S = \{(y,V,u) \in p^*E ~|~ u\in V\}\), i.e. this is just the subbundle of \(p^*E\) whose fiber over a point \((y,V)\in \Gr(d,E)\) is the vector space \(V\).
For convenience, let’s omit the \(y\) from the triplet defining a point in \(S\); instead we’ll just write a point as \((V, u)\) understanding that \(V\subset E_y\) is a \(d\)-dimensional subspace for some \(y\). We can define a map \[\varphi:\operatorname{Fr}(d,E)\times^{\GL_d}\mathbb C^d \to S\] by \[\big((v_1,...,v_d),(z_1,...,z_d)\big) \mapsto \left(\spann(v_1,...,v_d), \sum z_iv_i\right).\]
A \(g\in \GL_d\) acts on \((v_1,...,v_d)\) via \[(v_1,...,v_d)\cdot g = (u_1,...,u_d); ~ u_j = \sum_{i=1}^d g_{ij}v_i.\] Note here that if we want to think of \(v_1,...,v_d\) as living in \(E_y\), a \(n\)-dimensional vector space, we identify \(GL_d\) with the subgroup of \(GL_n\) fixing \(V = \spann(v_1,...,v_d)\). By this comment, \(\spann(v_1,...,v_d) = \spann(u_1,...,u_d)\), and thus
\begin{align*} \varphi:\big((v_1,...,v_d), (z_1,...,z_d)\big) &= \left(\spann(u_1,...,u_d), \sum_{i=1}^dz_iu_i\right) \\ &= \left(\spann(u_1,...,u_d), \sum_{i=1}^dz_i\sum_{\ell=1}^d g_{\ell i}v_i\right) \\ &= \left(\spann(u_1,...,u_d), \sum_{\ell=1}^d\Big(\sum_{i=1}^d g_{\ell i}z_i\Big)v_i\right) \\ &= \left(\spann(v_1,...,v_d), \sum_{\ell=1}^d\Big(\sum_{i=1}^d g_{\ell i}z_i\Big)v_i\right) \\ &= \varphi\big((v_1,...,v_d),g\cdot (z_1,...,z_d)\big), \end{align*}so the map \(\varphi\) is well-defined. It is surjective since for every point \((V, u)\in S\) we can find a basis \(v_1,...,v_d\) for \(V\) and a unique set of coefficients \(z_1,...,z_d\in \mathbb C\) such that \(u = z_1v_1 + ... + z_dv_d\), and it is injective since \(\GL_d\) acts simply transitively on the set of all ordered bases of \(V\).