The Lefschetz Theorem on Hyperplane Section

Theorem

If $M\subset {\mathbb{C}}^n$ is a non-singular affine algebraic variety on complex $n$-space with real dimension $2k$, then $H_i(M;Z)=0\text{for}i>k.$

Theorem

A complex analytic manifold $M$ of complex dimension $k$, bianalytically embedded as a closed subset of ${\mathbb{C}}^n$ has the homotopy type of a $k$-dimensional CW-complex.

Lefschetz

Let $V$ be an algebraic variety of complex dimension $k$ which lies in the complex projective space $\mathbb{C}{P}_n$. Let $P$ be a hyperplane in $\mathbb{C}{P}_n$ which contains the singular points (if any) of $V$. Then the inclusion map $V\cap P\to V$ induces isomorphisms of homology groups in dimensions less than $k-1$. Furthermore, the induced homomorphism $H_{k-1}(V\cap P;\mathbb{Z})\to H_{k-1}(V;\mathbb{Z})$ is onto.

Lefschetz

Under the hypothesis of the preceding corollary, the relative homotopy group ${\pi }_r(V,V\cap P)$ is $0$ for $r<k$.