In the following link$^{[1]}$, page $2$, we find the following question :

Let $X$ be a smooth $ \mathbb{Q} $ - variety and let $\mathcal{P} \mathrm{er} (X)$ be the subfield of $\mathbb{C}$ generated by the image of the pairing $(3)$. What is the transcendance degree of the finitely generated extension $ \mathcal{P} \mathrm{er} (X) / \mathbb{Q} $ ?

**My questions are** :

What is explitcitly, the definition of the set : $ \mathcal{P} \mathrm{er} (X) / \mathbb{Q} $ ? Is it defined by : $$ \mathcal{P} \mathrm{er} (X) / \mathbb{Q} = \left\{ \ P \left( \int_{ \gamma_{1} } \omega_1 , \dots , \int_{ \gamma_{m} } \omega_m \right) \;\middle|\; P \in \mathbb{Q} [t_1 , \dots , t_m ] \ \right\} $$ and if it's right, what is $m$ ? why ?

Why is $ \mathcal{P} \mathrm{er} (X) / \mathbb{Q} $ a finitely generated extension of $\mathbb{Q}$ ?

Thanks in advance for your help.

$^{[1]}$ *Periods and the conjectures of Grothendieck and Kontsevich–Zagier* by Joseph Ayoub (Universität Zürich, Switzerland) 2014