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Do we know any thing about cohomology of realization space of matroid (the space of all set of vectors in $\mathbb{C}^k$ which captures the independence structure of matroid $M$), more simple, for uniform matroid $U_{k,n}$? The realization space is totally determined by structure of matroid, so as the cohomology, can we write it? I found little paper about this thing, does people lost the interest to calculate it since the realization space can be vary complicated by Mnev's universal theorem?

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    $\begingroup$ Small comment: $U_{k,n}$ is usually represented as $n$ points general position in $\mathbb{C}^k$, even though you write $\mathbb{C}^n$. Maybe you have a different convention about what $k$ vs. $n$ means. (I would imagine this space, because of its "general position" nature, has not so bad cohomology compared to arbitrary matroid varieties, but don't know for sure.) $\endgroup$
    – Sam Hopkins
    Commented Nov 20, 2020 at 15:13
  • $\begingroup$ @SamHopkins, thanks for your comment, it's a typo, it's $\mathbb{C}^k$ $\endgroup$
    – J.D.Chern
    Commented Nov 22, 2020 at 6:53

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