The Grassmannian $Gr_1(\mathbb{C}^2)$ is another name for $\mathbb{P}^1$. If one endows $\mathbb{C}^2$ with a complex symplectic form, or if one prefers (since this will allow us to generalize in another direction), a complex volume form, i.e. a non-zero element of $\Omega \in \Lambda^2(\mathbb{C}^2)^*$, then given $\mathbf{a} := (a_0, a_1)^T \in \mathbb{C}^2 \setminus \{\mathbf{0}\}$ and coordinates $\mathbf{z} := (z_0,z_1)^T$ on $\mathbb{C}^2 \setminus \{ \mathbf{0} \}$, one may form the linear polynomial $$p(z_0, z_1) = \Omega(\mathbf{a}, \mathbf{z})$$ and may form higher degree polynomials by multiplying finitely many linear polynomials.
Now let us consider $Gr_2(\mathbb{C}^4)$ which is known to be biholomorphic to a smooth $4$-quadric in $\mathbb{P}^5$ via the Klein-Plücker embedding. Let $\Omega \in \Lambda^4(\mathbb{C}^4)^*$ be a non-zero element, so that it is a complex volume form on $\mathbb{C}^4$. Let $\mathbf{a}_1$ and $\mathbf{a}_2$ be two linearly independent vectors in $\mathbb{C}^4$, assumed to be fixed. On the other hand, let $\mathbf{z}_1$ and $\mathbf{z}_2$ be two linearly independent vectors in $\mathbb{C}^4$, which are assumed to be variable. Then define $$ p(\mathbf{z}_1, \mathbf{z}_2) = \Omega(\mathbf{a}_1, \mathbf{a}_2, \mathbf{z}_1, \mathbf{z}_2).$$ I would like to think of it as being analogue to the linear factors mentioned earlier. Thus it would make sense to talk about products of finitely many such factors. Note that the zero set of $p$ above consists of all linearly independent vectors $\mathbf{z}_1$ and $\mathbf{z}_2$ whose span intersects non-trivially the span of $\mathbf{a}_1$ and $\mathbf{a}_2$ (is that some kind of Schubert cycle by the way? I need to refresh my memory).
Guided by such analogies, it then makes sense to define the cross-ratio of $4$ sets of pairs of linearly independent vectors, say, $\mathbf{a}:=(\mathbf{a}_1, \mathbf{a}_2), \ldots, \mathbf{d}:=(\mathbf{d}_1, \mathbf{d}_2)$ by the formula $$ C(\mathbf{a},\mathbf{b};\mathbf{c},\mathbf{d}) = \frac{\Omega(\mathbf{a},\mathbf{d}) \Omega(\mathbf{b}, \mathbf{c})}{\Omega(\mathbf{a},\mathbf{c}) \Omega(\mathbf{b}, \mathbf{d})}.$$ Note that $C$ descends to a $4$-point function on $Gr_2(\mathbb{C}^4)$ (the $4$-quadric).
I just wonder whether such definitions, and the analogy with ordinary polynomials, were considered before. What are the properties of such a cross-ratio function? Maybe one may try to see whether some properties of polynomials carry over to such definitions. Of course, I could have asked similar things for $Gr_m(\mathbb{C}^{2m})$. The $p$s remind me a bit of the pairing between differential forms in mid dimension, which is important for example in homology and cohomology. Anyway, I am digressing. I was led to such questions by thinking about something rather different, namely the interplay between vectors and spinors. Thus for me, $\mathbb{C}^4$ is a space of Dirac spinors, associated to $4$-dimensional vectors.
I apologize that the post is not asking very specific questions. I am just curious if it will generate some interesting comments and/or answers.
