The complex projective plane $\mathbb{C}P^2$ can be thought of as the set of rank one 3-by-3 Hermitian matrices with norm one, i.e., $\mathbb{C}P^2 = \{xx^* : x \in \mathbb{C}^3, x^*x=1 \}$. As such, the space $\mathbb{C}P^2$ is a manifold with $4$-real dimensions embedded in the space of all $3$-by-$3$ Hermitian matrices, $H_3(\mathbb{C})$ (which has 9 real dimensions).

Suppose $V$ is an affine subspace of $H_3(\mathbb{C})$ with real dimension $7$, and that $V$ has nonempty intersection with $\mathbb{C}P^2$. Is the intersection simply connected, or is there a simple counterexample?