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.