Let $S/\mathbb{C}$ be a smooth quasi-projective variety and let $\mathbb{V}$ be a principally polarized variation of integral Hodge structures of pure weight $k$. Then we have a period map $$ S \to S_{\Gamma,G,M} = \Gamma \backslash G / M, $$ where $G$ is defined as so that $D = G/M$ is an open and closed part of $\mathbb{G}/\mathbb{P}$, where $\mathbb{G} \subseteq \mathrm{GL}(V_\mathbb{C})$ is the subgroup that preserves the principal polarization. Recall that we also defined a $\mathbb{R}_ \mathrm{alg}$-structure on the spaces $S_{\Gamma,G,M}$ with the property that Hecke correspondences are $\mathbb{R}_ \mathrm{alg}$-definable.
Theorem 1. The map $\Phi$ is $\mathbb{R}_ {\mathrm{an},\exp}$-definable.
What we will do is to find a nice compactification $S \subseteq \bar{S}$ so that $\bar{S}$ is smooth projective and $\bar{S} \setminus S$ is a normal crossing divisor. The question then becomes local on $S$, so that we can reduce to the case when $$ S = (\Delta^\ast)^r \times \Delta^{n-r} \subseteq \Delta^n = \bar{S}. $$
Theorem 2. The map $\Phi \colon (\Delta^\ast)^n \to S_{\Gamma,G,M}$ is $\mathbb{R}_ {\mathrm{an},\exp}$-definable.
In this case, the generators of $\pi_1((\Delta^\ast)^n) \cong \mathbb{Z}$ map to some $\gamma_i \in \Gamma$.
Theorem 3 (Borel). These elements $\gamma_i$ are quasi-unipotent, i.e., $\gamma_i^{m_i}$ is unipotent for some $m_i \ge 1$.
Proof.
The idea is to show that the conjugacy classes of $\gamma_i$ must have a limit point that lies in $M$. For $n = 1$, the map $\Delta^\ast \to \Gamma \backslash D$ lifts to the universal cover $$ \Phi^\prime \colon \mathcal{H} \to D. $$ On the $\mathcal{H}$ side, the generator is $ni \mapsto ni+1$ and therefore we have $$ \Phi^\prime(ni+1) = \gamma \Phi^\prime(ni). $$ On the $\mathcal{H}$ side, we have the metric $y^{-2}(dx^2 + dy^2)$, and on the $D$ side, we can find a metric that is invariant under $G$. It turns out that $\Phi$ doesn’t increase the metric, up to a scalar. Now if we write $g_n$ for a representative of $\Phi^\prime(ni)$ then we have $$ d(g_n^{-1} \gamma g_n M, M) = d(\gamma g_n M, g_n M) \le C d(in + 1, in) \le C/n. $$
So all this shows that there is a limit point in the conjugacy class of $\gamma$ that lies in $M$, and because $M$ is compact, we see that eigenvalues of $\gamma$ must have unit norm. Then because $\gamma \in \mathrm{GL}_ n(\mathbb{Q})$, we conclude that the eigenvalues must be roots of unity.
So now, we can change $(\Delta^\ast)^n$ by a finite étale cover and assume that all $\gamma_i$ are unipotent. Let us write $\gamma_i = \exp(N_i)$ where $N_i \in \mathfrak{g}$ are nilpotent.
Now we have $$ \begin{CD} \mathcal{H}^n @>{\tilde{\Phi}}>> D \br @VV{p = \exp}V @VVV \br (\Delta^\ast)^n @>{\Phi}>> S_{\Gamma,G,M} = \Gamma \backslash D. \end{CD} $$ Since we know the monodromy, we have $$ \tilde{\Phi}(z) = \exp\Bigl( \sum_{i=1}^n z_i N_i \Bigr) \psi(p(z)) $$ for some holomorphic $\psi \colon (\Delta^\ast)^n \to D$.
Theorem 4 (Schmid, nilpotent orbit theorem). The maps $\psi$ extend to holomorphic maps $\Delta^n \to D$.
So $\psi$ is $\mathbb{R}_ \mathrm{an}$-definable, because it is holomorphic. But we are not done yet, because $D \to S_{\Gamma,G,M}$ is not $\mathbb{R}_ {\mathrm{an},\exp}$-definable. So what we do is to look at the fundamental domains $$ S = \lbrace z \in h : \lvert \operatorname{Re}(z) \rvert \lt 1, \operatorname{Im}(z) \gt 1 \rbrace. $$ Then we need $\tilde{\Phi}(S^n)$ to be contained in a finite union of Siegel sets.
Theorem 5. For any $R \gt 0$ and $\eta \gt 0$, define the set $$ \mathcal{H}_ {R,\eta} = \lbrace z \in \mathcal{H} : \lvert \operatorname{Re}(z) \rvert \le R, \operatorname{Im}(z) \ge \eta \rbrace. $$ Then $\tilde{\Phi}(\mathcal{H}_ {R,\eta}^n)$ is contained in a finite union of Siegel sets.