TITLE: What is the fundamental group of a hypersurface?
QUESTION [5 upvotes]: (A related question is this 
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?

REPLY [4 votes]: I think that it is hard to say something is such a generality, as the following example shows.
Take $X= \mathbb{P}^1 \times \mathbb{P}^2$ and let $C \subset \mathbb{P}^2$ be a smooth curve of genus $g$. Set $Y = \mathbb{P}^1 \times C$. 
Then $\pi_1(Y) = \pi_1(C)$. The latter group is trivial for $g=0$, it is isomorphic to $\mathbb{Z} \oplus \mathbb{Z}$ for $g=1$ and it is a infinite nonabelian group for $g \geq 2$.