(Lefschetz hyperplane theorem)
Let \(X \subseteq \PP^N\) be a projective variety of dimension \(n\) over \(\CC\), and let \(Y = X \cap H\) be a hyperplane section such that \(X \setminus Y\) is smooth. Then the restriction maps \[ H^i(X, \ZZ) \to H^i(Y, \ZZ) \] are isomorphisms for \(i < n - 1\) and injective for \(i = n - 1\), and the maps \(\pi_i(Y) \to \pi_i(X)\) are isomorphisms for \(i < n - 1\) and surjective for \(i = n - 1\).
A smooth hypersurface \(Y \subseteq \PP^{n}\) with \(n \geq 3\) has \(\pi_1(Y) \cong \pi_1(\PP^{n}) = 1\) and \(H^2(Y, \ZZ) \cong H^2(\PP^{n}, \ZZ) \cong \ZZ\) when \(n \geq 4\). Iterating, a smooth complete intersection of dimension at least \(2\) in projective space is simply connected. A smooth plane curve of degree \(d \geq 3\) is a hyperplane section of dimension \(1\), and the theorem makes no claim about its fundamental group, which is nontrivial.
Every hypersurface section is a hyperplane section after a Veronese embedding, so the theorem applies to \(Y = X \cap V(F)\) for any homogeneous \(F\) with \(X \setminus Y\) smooth. The proof uses that \(X \setminus Y\) is an affine variety of dimension \(n\), which has the homotopy type of a CW complex of dimension at most \(n\).
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