Base subsets of polar Grassmannians

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Let Δ be a thick building of type Xₙ=Cₙ,Dₙ. Let also Gₖ be the Grassmannian of k-dimensional singular subspaces of the associated polar space Π (of rank n). We write Gₖ for the corresponding shadow space of type Xₙ,ₖ. Every bijective transformation of Gₖ sending base subsets to base subsets (the shadows of apartments) is a collineation of Gₖ, and it is induced by a collineation of Π if n≠ 4 or k≠ 1.