I am studying Voisin's Hodge Theory and Complex Algebraic Geometry, in order to better understand the underpinnings of her 2008 paper on Hodge structures.
She discusses elements $\omega$ in $H^{2}(X; \mathbf{R})$ that "polarize" the cohomology algebra $H^{\ast}(X;\mathbf{R})$ when $X$ is compact and Kaehler. She lists some properties of this "polarizing form/class", including the fact that the Lefschetz decomposition of the algebra is orthogonal with respect to the intersection form on $H^{k}(X;\mathbf{R})$ given by cupping $\omega^{n-k}$ with a pair of $k$-classes.
So far, though, I don't see an explicit definition for "polarizing form/class". Is this merely alternative terminology for the de Rham cohomology class of a Kaehler form? There is nothing to indicate that the properties she lists are, explicitly, the defining qualities of a "polarizing class". They just seem to be "useful things to know" about the class.
I would like to see if this polarizing concept can be extended to non-Kaehler forms, but first I need to understand it in the Kaehler case. Help, please?