  Let $G(k,n)$ be the complex Grassmann manifold of $k$-planes in $\mathbb
C^{k+n}$. In this note, we show that for $1<k<n$ and for any selfmap
$f:G(k,n)\to G(k,n)$, there exists a $k$-plane $V^k\in G(k,n)$ such that
$f(V^k)\cap V^k\ne \{0\}$.
