Let $X$ be a smooth compact projective manifold of dimension $n \geq 2$ which is simply connected. Take $D \subset X$ be a smooth divisor which is isomorphic to the disjoint union of $s>1$ copies of the projective space $\mathbb{P}^{d-1}$.
Is there an explicit nice expression for the fundamental group $\pi_1(X-D)$?
Intuitively I feel it should be some quotient of the free product of $s$ copies of $\mathbb{Z}$ but when I tried to prove it using the Van-Kampen theorem inductively I failed.