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?


1 Answer 1


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

  1. 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
  2. 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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.