For a compact Kaehler manifold $M$, a basic structural result for its de Rham cohomology is the hard Lefschetz theorem. See here or here for an overview of the result.
What happens in the non-compact case? Does the theorem fail, or is it just not understood?
Hard Lefschetz implies Poincaré duality with real coefficients, in particular $H^0\cong H^{2n}$ and for a non-compact connected manifold that never happens ($H^{2n}=0$).
For example: $M:=\mathbb{C}P^2\setminus\mathbb{C}P^1$ is contractible, but hard Lefschetz theorem (as stated) should have given an isomorphism between $H^0(M)=\mathbb{R}$ and $H^4(M)=0$.
UPD. Poincaré duality modification for non-compact manifolds (pointed out by BS.) suggests a Lefschetz-type map $\mu: H_c^{n-k}\to H^{n+k}$ defined by multiplication by $\omega^k$: $H_c^{n-k}\to H_c^{n+k}$ and subsequent composition with the canonical map from $H_c$ to $H$. $\mu$, hovewer, needs not be an isomorphism:
for $\mathbb{C}^2\setminus \{0\}$ Lefschetz map is identically zero, but $H^{3}=H_c^1$ are one-dimensional.
So, all in all, hard Lefschetz theorem fails for non-compact Kahler manifolds.
