periods of Mixed Hodge Structures

Question

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.

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.