Hyperbolicity and model-complete fields

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.