Algebraicity of analytic maps to a hyperbolic variety
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Let X be a complex algebraic variety. We say that X is Borel hyperbolic if, for every finite type reduced scheme S over the complex numbers, every holomorphic map from S to X is algebraic. We use a transcendental specialization technique to prove that X is Borel hyperbolic if and only if, for every smooth affine complex algebraic curve C, every holomorphic map from C to X 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.
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Mathematische Nachrichten, 293, 8, Wiley-VCH, Weinheim, 2020, https://doi.org/10.1002/mana.201900098