= Hard Lefschetz theorem on de Rham cohomology
{title2=$L^{n-k}:H^k(X;\mathbb R)\cong H^{2n-k}(X;\mathbb R)$}
For a compact <Kähler manifold> of complex dimension $n$, wedging with the $(n-k)$th power of its <Kähler class> is the displayed <isomorphism> for $0\leq k\leq n$. The <Hard Lefschetz isomorphism on Dolbeault cohomology> and <Hodge decomposition theorem for compact Kähler manifolds> give the result by summing the bidegrees. In particular, $L:H^k\to H^{k+2}$ is <injective> when $k<n$: if $L\alpha=0$, then $L^{n-k}\alpha=0$. Hence $b_k\leq b_{k+2}$ below the middle dimension.
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