Algebraicity of analytic maps to a hyperbolic variety

Abstract

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.

Non-archimedean hyperbolicity of the moduli space of curves

“…In the rest of this section we will prove the non-archimedean counterpart of the complex-analytic results in section 2.2 of [JK20]. After establishing those results about the rigid analytification, we shall give a proof of Theorem 3.1 following the line of reasoning in [JK20].…”

“…Motivated by conjectures of Green-Griffiths-Lang and higher dimensional generalizations of the Shafarevich problem, there has recently been much work on different notions of hyperbolicity [Lan86,Jav20] and the verification of them on the moduli spaces of smooth projective varieties [KL11,CP15,JSZ]. In this paper, we study the non-archimedean analogue of Borel hyperbolicity introduced in [JK20], and verify it for M…”

“…In this section we shall prove a non-archimedean analogue of Theorem 1.5 in [JK20]. Hereafter we denote X an as the rigid analytification of a finite type K-scheme X [Bo14, §5.4].…”

Arithmetic hyperbolicity and a stacky Chevalley–Weil theorem

“…Our goal in this section is to provide a precise interplay between the 'analytic' hyperbolicity of A g,C (that is, A an g,C is hyperbolically embedded in its Baily-Borel compactification) and the arithmetic hyperbolicity of A g,C (as proven by Faltings). We start with an analytic property of A an g,C (which is also studied in [26]). Lemma 6.2 (Borel's algebraization theorem).…”

“…Our goal in this section is to provide a precise interplay between the ‘analytic’ hyperbolicity of (that is, is hyperbolically embedded in its Baily–Borel compactification) and the arithmetic hyperbolicity of (as proven by Faltings). We start with an analytic property of (which is also studied in [26]). Lemma Let be a finite type reduced scheme over , and let be a morphism.…”