# What are the complex points of the real Grassmann variety?

If $X=Gr_{n,k}(\mathbb{R})$ is a real Grassmann variety (of $k$-planes in $n$-dimensional space), then what is $X(\mathbb{C})$, the set of complex points of $X$? In particular, can it be identified as a complex Grassmann variety?

If this is trivial and/or immediate from definitions, then a good reference for this material would be appreciated.

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Yes, it is a complex Grassmann variety, i.e., its points parameterize the $k$-planes in $\mathbb{C}^n$. In this holds for any field. This is not meant to be unhelpful, but...this is such a basic fact I'm not sure of what kind of reference to give. Maybe you need to say more about what definitions you're working from. – Pete L. Clark Jul 1 '11 at 21:18
The grassmanian is defined over $\mathbb Z$. – Mariano Suárez-Alvarez Jul 1 '11 at 21:25
@Pete L. Clark I knew this was an embarrassingly simple question to ask, but I felt more embarrassed not asking it. Basically, I've been working too long with too loose an understanding of algebraic geometry (variety$\approx$manifold sort of thinking) and would like to tighten up my understanding of it. Maybe I should have asked the question "What is a good resource for learning algebraic geometry?" – wckronholm Jul 1 '11 at 21:29

$@$wckronholm: right! I was thinking of coming back and adding this as answer, but I'm happy to see that you thought of it yourself. – Pete L. Clark Jul 2 '11 at 22:57