# periods of Mixed Hodge Structures

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: $\mathrm{Ext}^i(\mathbb{Q}(0),\mathbb{Q}(n))$?

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.
• 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. Commented Jan 19, 2015 at 6:55