# Hard Lefschetz in De Rham cohomology

I'm looking for a reference for Hard Lefschetz theorem in algebraic De Rham cohomology. By this I mean the statement that

If $i: Y \hookrightarrow X$ is a smooth hyperplane section of a smooth projective algebraic variety $X$ of dimension $n$ over a field $k$ of characteristic zero and $\omega \in H^2_{DR}(X)$ is its image under the cycle class map, then the operator

$L^j: H^{n-j}_{DR}(X) \to H^{n+j} _{DR}(X)$

sending $x$ to $x \cdot \omega^j$ is an isomorphism for $j=1, \ldots, n$.

Is a proof of this written somewhere?

Weak Lefschetz is proved in Hartshorne's paper "On the de Rham cohomology..." chapter III, section 7 but I was completely unable to find a reference for the former.

Thanks Francesco. Is that a proof for algebraic de Rham cohomology or the analytic one? I guess if you know how to do that for analytic you can deduce it for algebraic but you need some small argument when the base field is not $\CC$, don't you? – vicban Jun 21 '13 at 18:50
To expand Francesco's comment: after a field reduction followed an extension, you can assume that the ground field is $\mathbb{C}$. Apply Grothendieck's algebraic de Rham theorem to see that algebraic de Rham cohomology is isomorphic to singular cohomology. After this, the standard proof applies. However, if you are asking for a completely self contained elementary proof, then there isn't one. – Donu Arapura Jun 21 '13 at 18:57