On the simple connectedness of hyperplane complements in dual polar spaces

Let ΔΔ be a dual polar space of rank n≥4n≥4, HH be a hyperplane of ΔΔ and Γ≔Δ∖HΓ≔Δ∖H be the complement of HH in ΔΔ. We shall prove that, if all lines of ΔΔ have more than 3 points, then ΓΓ is simply connected. Then we show how this theorem can be exploited to prove that certain families of hyperplanes of dual polar spaces, or all hyperplanes of certain dual polar spaces, arise from embeddings.