Lefschetz hyperplane theorem is a relation between the homological invariants of an algebraic variety and its subvarieties.
\(X\subset \mathbb{C}\mathbb{P}^n\) be \(d\) dimensional complex projective variety and \(Y\) a hyperplanc section such that \(U=X-Y\) is smooth. In short, the Lefschetz hyperplane theorem says that \(Y\) contain enough homological information upto degree \(d-1\), precisely speaking,
\(H_k(Y,\mathbb{Z})\to H_k(X,\mathbb{Z})\) is isomorphism for \(k<d-1\) and surjective for \(k=d-1\).
Similar statement holds for \(\pi_k\).
\(H^k(X,\mathbb{Z})\to H^k(Y,\mathbb{Z})\) is isomorphism for \(k<d-1\) and injective for \(k=d-1\).
These are the same as saying the relative homology \(H_k(X,Y;\mathbb{Z})\), cohomology \(H^k(X,Y;\mathbb{Z})\) and homotopies \(\pi_k(X,Y)\) vanish to degree \(d-1\).