Is Hodge theory somehow connected with a Galois group action Gal(C/R)? I'm currently taking a course in Hodge theory ... and I wonder if all the splittings in $\{i,-i\}$ Eigenvalue pairs come from the Galois group action (of the extension $\mathbb{R}\rightarrow\mathbb{C}$) - it seems to me like that (and I couldn't find such a statement in my textbook).
Is this true? If yes, is this a good way to think of Hodge decomposition or does one need more data than just the Galois group? If not, what is my misconception?
I thought (if my assumption is true), this would be a way to generalize to other algebraic field extensions.. are there analogues of Hodge theory for any algebraic field extension? Does it involve the Galois group?
If this question isn't "researchy" enough, just close it ... I will come back asking questions in a year then :-)
 A: As far as I understand acc. to Gelfand/Manin "Homological Algebra" p. 140, the idea of Hodge structures comes from Galois representations in arithmetics and was then by Deligne's "yoga des poids" transfered to complex varieties. But there it is (or was? The book was written 20 years ago) unclear which symmetry group (called "Hodge symmetries" in the book) lurks behind it. 
A: You are correct: there is a connection to the Galois theory of $\mathbb{C}/\mathbb{R}$ here.
To give a Hodge structure on a real vector space $V$ -- i.e., a direct sum decomposition of its complexification into $(p,q)$ subspaces such that $H^{q,p}$ is the complex conjugate of $H^{p,q}$ -- is equivalent to giving an action of $G = \operatorname{Res}_{\mathbb{C}/\mathbb{R}} \mathbb{C}^{\times}$ on $V$.  Here 
$\operatorname{Res}_{\mathbb{C}/\mathbb{R}} \mathbb{C}^{\times}$ means the "restriction of scalars" from $\mathbb{C}/\mathbb{R}$ of the complex multiplicative group $\mathbb{C}^{\times}$.  In plainer terms, it means that we view $\mathbb{C}^{\times}$ not as a one-dimensional complex algebraic group, but as a 2-dimensional real algebraic group, a "nonsplit torus".  Then the fact that we have a homomorphism of real groups
$$G \to \operatorname{GL}(V)$$
implies an extra condition on the complexified representation
$\mathbb{C}^{\times} \to \operatorname{GL}(V \otimes \mathbb{C})$: namely that the space $V^{p,q}$ on which $z$ in $\mathbb{C}^{\times}$ acts 
as $z^{p} \overline{z}^q$ is the complex conjugate of the space $V^{q,p}$.
A brief (but accurate!) discussion of this can be found at
http://en.wikipedia.org/wiki/Hodge_structure#Hodge_structures
