Decomposition of Grassmannian

Question

Let $V$ and $W$ be two vector spaces of dimension $n$ each. Then is there a decomposition of the Grassmannian $G(r, V\oplus W)$ in terms of $G(r,V)$ and $G(r,W)$ ? I am expecting something similar to the wedge decomposition. My guess is that it is the direct sum of cartesian products of $G(k,V)$ and $G(l,W)$ such that $k+l=r$. Is this true ?

Answer 1

I don't think there is a reasonable way to construct $Gr(r,V\oplus W)$ from the $Gr(k,V)$'s and $Gr(l, W)$'s.
For example $Gr(1,\mathbb C\oplus \mathbb C)=\mathbb P^1(\mathbb C)$, a complex projective variety of dimension $1$.
But the $Gr(k,\mathbb C)$'s are sets with just $1$ element if $k=0$ or $k=1$ and $Gr(k,\mathbb C)=\emptyset$ for $k\geq 2$.
How on earth would you build $\mathbb P^1(\mathbb C)$ from pieces that are singleton sets or empty?