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Grassmannians are algebraic varieties whose points corresponds to vector subspaces of a fixed dimension in a fixed vector space.

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How few $k$-dimensional subspaces of $V$ are enough to have a complement to each $n-k$-dimen...

See these papers: Covering by Complements of Subspaces, II., W. Edwin Clark and Boris Shekhtman, Proc. Amer. Math. Soc.125 (1997), no. 1, 251--254. (link here, unrestriced access; MR review) Coverin …
W. Edwin Clark's user avatar