$\DeclareMathOperator\Gr{Gr}$Consider $\mathbb{C}^n$ endowed with the Hermitian inner product $\langle u,v\rangle=u^*v$, and let $U \subseteq \Gr(k,\mathbb{C}^n)$ be a Zariski open dense subset of the Grassmannian of $k$ planes in $\mathbb{C}^n$. Is the set \begin{align} V=\{u^{\perp} | u \in U\}\subseteq \Gr(n-k,\mathbb{C}^n) \end{align} of orthogonal complements (under $\langle\cdot,\cdot\rangle$) open dense in $\Gr(n-k,\mathbb{C}^n)$? Or does it at least contain an open dense subset of $\Gr(n-k,\mathbb{C}^n)$?

If the bijection $\Gr(k,\mathbb{C}^n)\leftrightarrow \Gr(n-k,\mathbb{C}^n)$ given by $u \leftrightarrow u^\perp$ were an isomorphism of algebraic varieties then this would be obvious, but unfortunately it appears to only be an isomorphism when these are viewed as varieties over the reals.

Another idea is to somehow use Chevalley's theorem, although this result doesn't seem to hold over the reals.