  Let $X$ be an algebraic variety over $\mathbb{C}$. We say that $X$ is Borel
hyperbolic if, for every finite type reduced scheme $S$ over $\mathbb{C}$,
every holomorphic map $S^{an}\to X^{an}$ is algebraic. We use a transcendental
specialization technique to prove that $X$ is Borel hyperbolic if and only if,
for every smooth affine curve $C$ over $\mathbb{C}$, every holomorphic map
$C^{an}\to X^{an}$ is algebraic. We use the latter result to prove that Borel
hyperbolicity shares many common features with other notions of hyperbolicity
such as Kobayashi hyperbolicity.
