Put $V=\{A\in H_3(\mathbb{C}):a_{11}=a_{33}=1/3\}$. I claim that $V\cap\mathbb{C}P^2$ is the set of matrices of the form $$ P = \frac{1}{3}\left[\begin{array}{ccc} 1 & z & zw \\ \overline{z} & 1 & w \\ \overline{zw} & \overline{w} & 1 \end{array}\right], $$ where $|z|=|w|=1$. The proof is not hard if you just expand out the equations $P^2=P^*=P$ and $\text{trace}(P)=1$, looking at the diagonal entries first. We conclude that $P$ is a torus, and so is not simply connected.
In terms of the more usual picture of $\mathbb{C}P^2$, we just have $$ P = \{[x_0:x_1:x_2] : |x_0|=|x_1|=|x_2|\}. $$