What can one say about the relation between the Hodge decompositions 
of $H^\*(X,C)$ and $H^{*}(X_\sigma,C)$
for a complex algebraic smooth projective variety $X$ and $\sigma$ an automorphism
of the field of complex numbers ? I am also interested in conjectural results, e.g.
whether Standard Conjectures imply something about the relationship of
the Hodge structure of conjugate varieties $X$ and $X_\sigma$.


 For example, it appears that the Hodge numbers $h^{ab}=dim_C H^a(X,\Omega_X^b)$
and $h_\sigma ^{ab}=dim_C H^a(X_\sigma,\Omega_{X_\sigma}^b)$ are necessarily
the same (being defined via de Rham cohomology); what other invariants/subspaces etc are the same or known to be possibly different ? 
Can one define a related Galois (or $Aut(C/Q)$) action on the Hodge structures,
something like where an automorphism $\sigma\in Aut(C/Q)$ takes the Hodge structure of $X$ 
into that of $X_\sigma$'s ? 


I know of several references that show that conjugate varieties
may be rather different topologically. F.Charles, <a href="http://arXiv.org/abs/0706.3674v3">
Conjugate varieties with distinct real cohomology algebras</a>
lists most references I know of and well, proves what's in the title.
Serre constructed two two non-homotopic conjugated varieties with different fundamental groups;
his proof exploits the fact that for a CM elliptic curve, its fundamental group has different
$EndE$-module structures for different embeddings. <a href="http://www.maths.ed.ac.uk/~aar/topology/Topology%20Vol%2013/Abelson_Topologically-distinct-conjugate-varieties-with-finite-fundamental-group_1974.pdf">Abelson</a> constructs two non-homotopic conjugated varieties with the same finite fundamental groups.