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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
up vote 4 down vote accepted

OK. After a little thought, I've realized how basic my question is. The equations defining the real variety are exactly the same as the ones which define the complex variety, and define the corresponding variety for any field.

(I'm not really sure any more what I was confused about in the first place.)

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$@$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

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