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.
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.)