Value Distributions

Abstract

We fix n, and for each k, 0 ≤ k ≤ n, consider ∧^{k +1} C ^{n +1}. Set

$$n(k) = (\begin{array}{*{20}{c}} {n + 1} \\ {k + 1} \end{array}) - 1$$

so that ∧^{k +1} C ^{n +1} ≃ C ^{n (k)+1} Let G (n, k) be the Grassmannian of k-planes in P _n C, i.e., the Grassmannian of (k + 1)-dimensional subspaces in C ^{n +1} Then dim G (n, k) = (n − k)(k + 1). To a (k + 1)-dimensional subspace spanned by a ₀, …, a _k ∈ C ^{n +1}, we assign a decomposable (k + 1)-vector A = a ₀ ∧ … ∧ a _k ∈ ∧^{k +1} C ^{n +1} which is determined, up to a constant factor, by the subspace. Conversely, each decomposable (k + 1)-vector A determines a k-plane in P _n C, i.e., a (k + 1)-dimensional vector subspace of C ^{n +1} both of which will be denoted by the same symbol [A]. This correspondence defines the Plücker imbedding

$$G(n,k) \subset {P_{n(k)}}C.$$

(8.1.1)