The following trick is folklore: choosing embeddings, a birational map, and whatever else might be necessary to witness that $X_{\mathbb{C}}$ is rational, the conditions expressing this fact (for these particular data witnessing it!) are finitely many polynomial equations, with algebraic coefficients, on finitely many complex numbers. But a system of polynomial equations over $\bar{\mathbb{Q}}$ which has a complex solution has one over $\bar{\mathbb{Q}}$ (e.g., by Nullstellensatz), and this algebraic solution witnesses rationality of $X$. ∎ The same idea works for many other properties.
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