Q: what is the definition of Hodge structure of geometric origin Let $H$ be a mixed Hodge structure or, more generally, a mixed Hodge structure over a subfield $k$ of $\mathbb{C}$, by which I mean a $k$-vector space with two filtrations (Hodge and weight), a $\mathbb{Q}$-vector space with a filtration (weight) and an isomorphism between the complexifications such that weight filtrations correspond to each other and these data induce a mixed Hodge structure on the $\mathbb{Q}$-vector space. 
The main example is given by the cohomology of an algebraic variety over $k$. This and all their sub Hodge structures certainly have geometric origin. But I guess there should be more, for example I want the cohomology of a variety with values in the local system $R^n f_\ast \mathbb{Q}$ for some morphism of varieties $f \colon X \to Y$ to be of geometric origin as well. 
What is the general definition? Is there common agreement about it? 

A: To answer your last question, I don't think there is a commonly accepted definition. So let me propose a couple right now.


*

*Let $N$ be Nori's category of mixed motives over $k$. There is now (or soon to be) a book by Huber and Müller-Stach that gives a reasonably complete account of this. There is a realization functor from $N$ the category of mixed Hodge structures. Take the essential image. This includes the mixed Hodge structures $H^i(X, R^jf_*\mathbb{Q})$ for $f$ projective by  The Leray spectral sequence is motivic 

*If you don't want to think about motives, then you can take the largest subcategories of the derived category of mixed Hodge modules on each $X$, containing constant modules $\mathbb{Q}_X^H$, and stable under direct images, inverse images, etc. The heart of this category over a point is what you would want.


The category given in first definition would lie in  the second but I don't know if they coincide.
