This is a naive question.

One is told that, somehow, Hodge theory for varieties over complex numbers, is an analog of weight theory for varities over finite fields. In weight theory, one considers eigenvalues of Frobenius and so on. Hodge theory should capture symmetries of the Galois group of complex numbers over real numbers.

My question is: It is confusing; For a variety over a finite field, it makes sense that the Galois group will act on different things and give some information etc.; But in Hodge theory, we begin with a variety over the complex numbers, not over the real numbers; So what is the logic by which the Galois group of complex numbers over real numbers enters the picture? What is the logic behind the words "hidden symmetries" in the book by Gelfand and Manin?

i.e., when we have a variety over the algebraic closure of a finite field, we do not claim to have "weight theory"; We need structures to be defines over the finite field..

I hope my naive question is clear,

Thank you, Sasha

definition(so the Galois group of $\mathbb{C}/\mathbb{R}$ does not really enter the picture). The structure comes from the field ofcoefficients: You take cohomology with coefficients in $\mathbb{Q}$ (or $\mathbb{Z}$) and then extend scalars to $\mathbb{C}$ (remembering the $\mathbb{Q}$-structure). If you would start with $\mathbb{C}$-coefficients, there is no way to get the Hodge structure. Hope this small comment clarifies the confusion a bit. $\endgroup$ – jmc Nov 12 '13 at 9:31algebraic de Rhamcohomology (which is comparable with singular cohomology and its Hodge structure), and$p$-adic Hodge theory(involving the $p$-adic cohomology, where $p$ is the residue characteristic). $\endgroup$ – jmc Nov 20 '13 at 10:22