Given a geometrically connected variety $X$ over $\mathbb{Q}$ we have a short exact sequence $$ 1\to \pi_1(X_{\overline{\mathbb{Q}}})\to \pi_1(X)\to Gal(\overline{\mathbb{Q}}/\mathbb{Q})\to 1. $$ A rational point on $X$ provides a splitting of this sequence i.e. it exhibits $\pi_1(X)$ as a semidirect product of $\pi_1(X_{\overline{\mathbb{Q}}})$ and the absolute Galois group of $\mathbb{Q}$. Does there exist a smooth and proper variety (to avoid trivialities with an infinite geometric fundamental group) such that $\pi_1(X)$ is actually the direct product of these two groups? Is there an example where such a splitting comes from a rational point?

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