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arXiv · 2403.15300

Hyperbolicity and model-complete fields

Michał Szachniewicz·Jinhe Ye
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Abstract

We study model-complete fields that avoid a given quasi-project variety $V$. There is a close connection between hyperbolicity of $V$ and the existence of the model companion for the theory of characteristic-zero fields avoiding rational points on $V$. This gives a model theoretic notion of hyperbolicity that we call excludability. In particular, we show that if $V$ is a Brody hyperbolic projective variety over $\mathbb{Q}$ with $V(\mathbb{Q}) = \varnothing$, then the model companion, called $V\XF$, exists. We also study some model-theoretic properties of $V\mathrm{XF}$. This extends the results for curves by Will Johnson and the second author.

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Michał Szachniewicz, Jinhe Ye. 2024-03-22. Hyperbolicity and model-complete fields. https://doi.org/10.1093/imrn%2Frnaf019

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