For a compact Kaehler manifold $M$, a basic structural result for its de Rham cohomology is the hard Lefschetz theorem. See [here](https://en.wikipedia.org/wiki/Lefschetz_hyperplane_theorem#Hard_Lefschetz_theorem) or [here](https://ncatlab.org/nlab/show/hard+Lefschetz+theorem) for an overview of the result. 


What happens in the non-compact case? Does the theorem fail, or is it just not understood?



Non-compact hard Lefschetz theorem
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.


