Maps between certain complex Grassmann manifolds

Let k , l , m , n be positive integers such that m − l ≥ l > k , m − l > n − k ≥ k and m − l ≥ 2 k 2 − k − 1 . Let G k ( C n ) denote the Grassmann manifold of k-dimensional vector subspaces of C n . We show that any continuous map f : G l ( C m ) → G k ( C n ) is rationally null-homotopic. As an application, we show the existence of a point A ∈ G l ( C m ) such that the vector space f ( A ) is contained in A; here C n is regarded as a vector subspace of C m ≅ C n ⊕ C m − n .