# Stratify Grassmannian based on dimension of intersection

Consider a vector space $$V$$ of dimension $$n$$ and an automorphism $$A:V\rightarrow V$$. Using this we can define, $$\Sigma_i = \{W\in Gr_k V| \dim(W\cap AW) = i\}$$ for $$i=0,\ldots, k$$.

My question is what can we say about these $$\Sigma_i$$? They seem like singular varieties. Can we find out the dimensions? Do we have a stratification of $$Gr_kV$$?

Clearly the dimension should depend on properties of $$A$$. For example, if $$A=Id$$ then $$\Sigma_k= Gr_k V$$ and all other $$\Sigma_i=\emptyset$$. In general, my guess is that any result should depend on the Jordan decomposition of $$A$$.

Any help regarding this appreciated.