TITLE: On the fundamental group of hypersurfaces
QUESTION [6 upvotes]: Let $H$ be a smooth projective hypersurface in $\mathbb{P}^n(\mathbb{C})$ where $n\geq 3$. Then by the Lefschetz hyperplane theorem we have that $H^1(H,\mathbb{C})=
H^1(\mathbb{P}^n(\mathbb{C}),\mathbb{C})=0$. It thus follows that $\pi_1(H)^{ab}$ is a finite
abelian group.
Q.1 Do we have an example of an $H$ such that $\pi_1(H)$ is infinite?
Q.2 Is it possible to compute explicitly $\pi_1(H)$ (or more modestly 
$\pi_1(H)^{ab})$ in term of the defining equation of
$H$ ?

REPLY [7 votes]: Let me supplement Algori's answer a bit. The statement for fundamental groups goes
back to Zariski, I believe.  Standard Morse theory proofs yield this and more ( Milnor's
book gives a nice account).
Over any field, there are similar results using the etale fundamental group.
See for example SGA2, exp XII, cor. 3.5.