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.

Help please! ;)