The Polarization of Hodge structure is defined as a bilinear form $S$ on rational Hodge cohomology $H_{\mathbb{Q}}$ of given n dimensional projective variety $X$, more explicitly, defined by the Hodge-Riemann bilinear form, say, for $\alpha \in H^{p, q}_{\mathbb{Q}}$ and $\beta \in H^{q, p}_{\mathbb{Q}}$
$S(\alpha, \beta)=\int_{X}\alpha\wedge\beta\wedge \omega^{n-2(p+q)}$
Here $\omega$ the kahler form of $X$ .
I wonder why we call this bilinear form a polarization? We name algebraic variety equipped with a positive line bundle as polarized variety, because the line bundle here give a linearlization, say if we have a group action on based variety, we can lift it to the line bundle as a linear action, just like the polar coordinate. But how about the case of polarization of Hodge structure?
Thank you for your answer.