Lefschetz hyperplane theorem

Remarks

The idea of the proof: $X \setminus D$ is a smooth affine variety of dimension $n$. By the Andreotti–Frankel theorem it has the homotopy type of a CW complex of real dimension at most $n$. Lefschetz duality then gives $H^i(X, D; \Z) \cong H_{2n - i}(X \setminus D; \Z) = 0$ for $i < n$, and the long exact sequence of the pair gives the claim. The homotopy version says that $\pi_i(D) \to \pi_i(X)$ is an isomorphism for $i \le n - 2$ and surjective for $i = n - 1$.

Example: a smooth hypersurface $X \subseteq \mathbb{P}^{n+1}_\C$ of dimension $n$ is an ample divisor in $\mathbb{P}^{n+1}$. So $H^i(X, \Z) \cong H^i(\mathbb{P}^{n+1}, \Z)$ for $i \le n - 1$, which is $\Z$ for even $i$ and $0$ for odd $i$; in particular $X$ is simply connected when $n \ge 2$.

The bound $n - 1$ is sharp. Let $X = \mathbb{P}^1 \times \mathbb{P}^1$, of dimension $n = 2$, embedded in $\mathbb{P}^3$ as a quadric by the Segre embedding. A hyperplane section is a smooth conic $D \cong \mathbb{P}^1$, and $H^2(X, \Z) \cong \Z^2$ while $H^2(D, \Z) \cong \Z$. So the map in degree $n = 2$ is not injective.

Going deeper

For a student

The theorem is the basis for induction on dimension in the topology of algebraic varieties, through Lefschetz pencils. Together with Poincaré duality it shows that all the interesting cohomology of a smooth complete intersection sits in the middle degree. It is complemented by the hard Lefschetz theorem, which describes the middle and upper degrees, and it has analogues in étale cohomology used in the proof of the Weil conjectures.