Two Questions:

First. As I know the notion of periods comes when one has two vector spaces over a subfield $k$ of $\mathbb{C}$ (usually given by two cohomology theories) and an isomorphism between their extensions to $\mathbb{C}$. But I'm confused when this notion is used for a mixed Hodge structure. What are two vector spaces over $\mathbb{Q}$ and the isomorphism between their tensor with $\mathbb{C}$? Or we need additional data for attaching periods to a MHS?

What is the geometric meaning of periods for cohomology groups of a variety over $\mathbb{C}$?

Second. What is the relation between periods of MHS's and Ext groups: $\mathrm{Ext}^i(\mathbb{Q}(0),\mathbb{Q}(n))$?

Thanks for your answers and references suggestions.

Edit. I found a reference: Goncharov, Volumes of hyperbolic manifolds and mixed Tate motives for the definition of periods for mixed Hodge-Tate structures. It seems that the definition only works for Hodge-Tate structures and cannot be generalized for all MHS's. But the question of "geometric meaning" also remains in this case.


1 Answer 1


The classical notion of period, is the (set of) integral(s) of an algebraic differential form along cycles on a variety. As you said yourself, this can be expressed as an entry of the change of basis matrix from algebraic de Rham to singular (or Betti) cohomology. This still makes sense for a singular or open variety. E.g. if $X$ is smooth variety, then we can find a smooth proper compactification $\bar X$ such that $X= \bar X-D$, where $D$ is a divisor with normal crossings. So we have, by Deligne, a comparison map $$H^*(X,\mathbb{Z})\otimes \mathbb{C}\cong H^*(\bar X,\Omega_{\bar X}(\log D))$$ So you get a period matrix as above (after choosing bases).

For the second, I am not entirely sure what "period" means either. However, there should certainly be a map from $Ext^*$ in any reasonable category of mixed motives, to $Ext^*$ in the category of polarizable MHS. But note that $Ext^i=0$ for $i>1$ in PMHS (Beilinson), so there is no information in this case.

  • $\begingroup$ Thanks! So does it makes sense to talk about the periods of a mixed Hodge structure? I think for defining periods we need a notion of de Rham cohomology over $\mathbb{Q}$ (or a larger extension) which is not present just in the MHS of a variety. $\endgroup$
    – Mostafa
    Commented Jan 19, 2015 at 6:55

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.