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.

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.