Let $X$ be a compact Kähler manifold of complex dimension $n$, oriented by its complex structure, and let $[X]\in H_{2n}(X;\mathbb Z)$ denote its fundamental class. Let $[\omega]\in H^2(X;\mathbb R)$ be the cohomology class of a Kähler form, viewed also as an element of $H^{1,1}(X)\cap H^2(X;\mathbb R)\subset H^2(X;\mathbb C)$. For each integer $m$, define the real Lefschetz operator
and let the same symbol denote its complex-bilinear extension to $P^k(X;\mathbb C)\times P^k(X;\mathbb C)$. Then $P^k(X;\mathbb R)$ is a real pure Hodge structure of weight $k$, with
If $[\omega]$ lies in the image of the natural map $H^2(X;\mathbb Q)\to H^2(X;\mathbb R)$, choose a rational class $\omega_{\mathbb Q}\in H^2(X;\mathbb Q)$ whose image is $[\omega]$. Define the rational Lefschetz operator $L_{\omega_{\mathbb Q}}:H^m(X;\mathbb Q)\to H^{m+2}(X;\mathbb Q)$ by $L_{\omega_{\mathbb Q}}(\gamma)=\omega_{\mathbb Q}\smile\gamma$, and define