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 obtained from homotopy equivalent to $V$ by gluing with cells of dimension $\dim_{\mathbf{C}}M$ or greater\geq \dim_{\mathbf{C}}M$glued to it. 1 Non-singular projective hypersurfaces are simply connected. By the Lefschetz theorem$\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$smooth)$M$is obtained from$V$by gluing cells of dimension$\dim_{\mathbf{C}}M\$ or greater.