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
$H$ ?



On the fundamental group of hypersurfaces
Non-singular projective hypersurfaces are simply connected. By the Lefschetz theorem $\pi\_k(X)\to\pi\_k(\mathbf{P}^n(\mathbf{C}))$ is an isomorphism for $k\leq n-2$ where $X$ is a nonsingular complex hypersurface: as shown e.g. in Griffiths-Harris (chapter 1, second proof of the Lefschetz hyperplane theorem) if $M$ is a smooth complex manifold and $V$ is the zero locus of a section of a positive line bundle, then (assuming $V$ smooth) there is a smooth function on $M$ with $V$ as the zero locus and all critical points outside $V$ non-degenerate and of index $\geq \dim\_{\mathbf{C}}M$; so $M$ is homotopy equivalent to $V$ with cells of dimension $\geq \dim\_{\mathbf{C}}M$ glued to it.


