Hard Lefschetz Theorem

Let $X$ be a Kähler manifold of complex dimension $dim_{\mathbb C} = n$. Let $[\omega]$ be the cohomology class of a Kähler metric on $X$. Then powers of the class $[\omega]$ define a linear morphism between groups $$ L^k: H^{n-k}(X, {\mathbb C}) \longrightarrow H^{n+k}(X, \mathbb C)$$ Hard Lefschetz asserts that this is, in fact, an isomorphism of vector spaces. Proof to come soon!