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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.

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