Maps between certain complex Grassmann manifolds

Let k,l,m,nk,l,m,n be positive integers such that m−l≥l>km−l≥l>k, m−l>n−k≥km−l>n−k≥k and m−l≥2k2−k−1m−l≥2k2−k−1. Let Gk(Cn)Gk(Cn) denote the Grassmann manifold of k  -dimensional vector subspaces of CnCn. We show that any continuous map f:Gl(Cm)→Gk(Cn)f:Gl(Cm)→Gk(Cn) is rationally null-homotopic. As an application, we show the existence of a point A∈Gl(Cm)A∈Gl(Cm) such that the vector space f(A)f(A) is contained in A  ; here CnCn is regarded as a vector subspace of Cm≅Cn⊕Cm−nCm≅Cn⊕Cm−n.