(Hard Lefschetz)
Let \(X\) be a smooth projective complex variety of dimension \(n\), and let \(\omega \in H^2(X, \QQ)\) be the class of a hyperplane section. For every \(0 \leq k \leq n\), cup product with \(\omega^k\) is an isomorphism \[ L^k \colon H^{n-k}(X, \QQ) \xrightarrow{\ \sim\ } H^{n+k}(X, \QQ) . \]
The Betti numbers satisfy \(b_{i-2} \leq b_i\) for \(i \leq n\), and \(H^\bullet(X, \QQ)\) has the Lefschetz decomposition into primitive classes \(P^{n-k} = \ker\big(L^{k+1} \colon H^{n-k} \to H^{n+k+2}\big)\): \[ H^m(X, \QQ) = \bigoplus_{j \geq 0} L^j P^{m-2j} . \]
The weak Lefschetz theorem is the Lefschetz hyperplane theorem T-LEFHYP: restriction to a smooth hyperplane section \(Y\) is an isomorphism \(H^i(X) \to H^i(Y)\) for \(i < n-1\) and injective for \(i = n-1\). The hard Lefschetz theorem fails for compact complex manifolds that are not Kähler, and for singular projective varieties unless ordinary cohomology is replaced by intersection cohomology. On a smooth projective curve it says \(H^0 \cong H^2\) via the class of a point; on a surface it says \(H^1 \cong H^3\) via a hyperplane class, so \(b_1 = b_3\), which Poincaré duality also gives.