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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:

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

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

  3. 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).

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

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    $\begingroup$ Learn about complex-analytic projective geometry, since that is what provides the examples that motivate the abstract definitions. Also see Deligne's papers on Hodge structures. $\endgroup$ – nfdc23 Oct 9 '17 at 15:07
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    $\begingroup$ @nfdc23 What specifically? Do you have a helpful suggestion for a particular topic/construction which can motivate the definition of a polarization? $\endgroup$ – Saal Hardali Oct 9 '17 at 15:21
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    $\begingroup$ You'll find all the answers to your question in Hodge theory and Complex algebraic geometry by C. Voisin. $\endgroup$ – abx Oct 9 '17 at 16:29
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    $\begingroup$ @abx That's exactly the one I'm reading right now. Can you give a specific reference inside that discusses motivation for polarization? What is suggested there is that the kahler form itself provides a polarization, I can't find anywhere motivation for defining this thing however, why not just define a something which is hodge structure together with a leftchetz operator for instance? $\endgroup$ – Saal Hardali Oct 9 '17 at 16:58
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    $\begingroup$ And forget the cup-product? It is fundamental in algebraic geometry, already for curves (giving the principal polarization of the Jacobian), for surfaces (look at any text on K3 surfaces), etc. $\endgroup$ – abx Oct 9 '17 at 17:30
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Perhaps it's not obvious at first, but the notion of polarization is important. Some examples, where it comes up:

  1. A complex torus $X=\mathbb{C}^n/L$, where $L$ is a lattice, is compact Kähler, so $H^1(X,\mathbb{Z})$ carries a Hodge structure. A classical theorem of Riemann can be understood as saying that $X$ is an abelian variety (i.e. $X$ "comes from" algebraic geometry) if and only if this Hodge structure carries a polarization.
  2. If $L$ is a local system underlying a polarizable variation of rational Hodge structure, whatever that means, over a quasiprojective base then the monodromy representation is semisimple. In this generality, the result is due to Schmid and polarizations are essential.
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