(A related question is this 
https://mathoverflow.net/questions/51301/on-the-fundamental-group-of-hypersurfaces).

Let $X$ be a simply connected projective complex manifold of dimension at least 3. 
Let $Y\subset X$ be a smooth hypersurface. What is $\pi_1(Y)$ then?
If $Y$ is a hyperplane section, then $\pi_2(X,Y)=0$ by the Lefschetz
theorem, hence $\pi_1(Y)=0$. The question is, can we say anything about the fundamental 
group of a hypersurface without assuming ampleness?