Maps between certain complex Grassmann manifolds

Abstract

Let $k,l,m,n$ be positive integers such that $m-l\ge l>k, m-l>n-k\ge k$ and $m-l>2k^2-k-1$. Let $G_{k}(\mathbb{C}^n)$ denote the Grassmann manifold of $k$-dimensional vector subspaces of $\bc^n$. We show that any continuous map $f:G_{l}(\bc^m)\to G_{k}(\mathbb{C}^n)$ is rationally null-homotopic. As an application, we show the existence of a point $A\in G_{l}(\bc^m)$ such that the vector space $f(A)$ is contained in $A$; here $\mathbb{C}^n$ is regarded as a vector subspace of $\mathbb{C}^m\cong \bc^n\oplus\bc^{m-n}.$