A Hodge structure can be defined as a real, algebraic representation of the Deligne torus ${Res}^\mathbb{C}_{\mathbb{R}}\mathbb{G}_m$. Coming from Kahler manifolds the intuition for this is clear. The complex structure on the smooth tangent bundle of a Kahler manifold is itself a sort of parametrized hodge structure (of weight -1) and so are the exterior powers of the cotangent bundle.

For the definition of a polarized Hodge structures however I can find almost no intuition. All the definitions I saw so far seem to suggest that a polarization is a sort of bilinear form inside the category of hodge structures with some "extra conditions". I won't pronounce the definition since I can't say I have a specific problem with anything, I only wish there was a way to motivate it.

For instance:

Is the polarization related to the intersection form coming from ordinary cohomology? If so how?

What is polarization good for? Why isn't the intersection form enough? Does the polarization carry more information? If so what kind?

Is there a canonical way to define it similar to the definition of hodge structure I gave above? Ideally such a way would make all the sign issues in this definition transparent, and would be invariant to the convention one uses for the tate twist $\mathbb{R}(n)$. (whether you twist by $2\pi i$ or not).

Do polarization come up in $l$-adic setting? If so how?

Hodge theory and Complex algebraic geometryby C. Voisin. $\endgroup$ – abx Oct 9 '17 at 16:29