While working on a problem of differential topology I stumbled on a question of algebraic geometry that seems pretty basic, but that I'm unable to answer because I know very little algebraic geometry. I hope somebody could help:

It's known that if $X$ and $Y$ are (algebraic) subvarieties of $d$-dimensional (complex) projective space $P^d$ with complementary dimensions (i.e. $\dim X + \dim Y = d$) then $X \cap Y \neq \emptyset$ (in particular, any two curves in the projective plane $P^2$ always intersect).

My question is: Does this theorem hold if $P^d$ is replaced by the Grassmannian $G=G(m,n)$ (the set of $m$-dimensional subspaces of $\mathbb{C}^n$)?

In other words: If $X$, $Y$ are subvarieties of $G=G(m,n)$ whose dimensions sum to $\dim G(m,n) = m(n-m)$, does it follow that they have non-empty intersection?

In fact, in the situation I'm interested in, $Y$ is just a immersed smaller grassmannian $G(m-k,n-k)$. [More precisely, $Y$ is the set of $m$-dimensional subspaces of $\mathbb{C}^n$ that contain a given fixed $k$-dimensional subspace, where $k < m < n$.]

If the answer to the question is no (even for this particular $Y$), I pose a sharper question: What is the least $s=s(m,n)$ such that for any subvariety $X$ of $G$, if $\dim X + \dim Y \geq s$ (where $Y$ is as above) then $X \cap Y \neq \emptyset$?

Thanks for any help!