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 $H$ ?