Correction to: Demailly’s notion of algebraic hyperbolicity: geometricity, boundedness, moduli of maps (Mathematische Zeitschrift, (2020), 296, 3-4, (1645-1672), 10.1007/s00209-020-02489-6)

The proofs of Theorem 1.14 in [1] and of its “pseudofied” extension, Theorem 1.26 in [2], contain a gap in the argument. This mistake does not affect any of the other results nor proofs in the rest of the papers. Let X be a bounded projective variety over an algebraically closed field k of characteristic zero. By definition, for every ample line bundle L and smooth projective curve C and every nonconstant morphism f:C→X, the boundedness of X implies the existence of a bound on degf∗L depending only on X, L, and C. Theorem 1.14 of [1] asserts that one can bound degf∗L uniformly in the genus of C. More precisely, given a projective bounded variety X, for every integer g and every ample line bundle L on X, there exists a real number α(X,L,g) such that, for every smooth projective curve C of genus g and every non-constant morphism f:C→X, one has (Formula presented.) Let us explain the mistake in the proof of Theorem 1.14 of [1] (the same issue occurs in the proof of Theorem 1.26 of [2]). In the notation of that proof, one first shows that the Hom-scheme Homk(Y,X×Z) is of finite type over k. Our proof then asserts, without justification, that the Hom-scheme HomZ(Y,X×Z) is also of finite type. This statement is likely true but requires a proof. Consider the natural morphism of Hom-schemes (Formula presented.) where πZ:X×Z→Z is the projection. In the original argument we incorrectly identified HomZ(Y,X×Z) with the scheme-theoretic fibre of comp over the structure morphism Y→Z (viewed as a k-point of Homk(Y,Z)). Although this fibre is closed in Homk(Y,X×Z), and therefore of finite type, it cannot be identified with HomZ(Y,X×Z) in any natural sense. (Rather, it should be viewed as the Weil restriction of HomZ(Y,X×Z) from Z to Speck.) We do not know how to repair this proof. Theorem 1.14 in [1] (resp. Theorem 1.26 in [2]) is thus a uniform boundedness statement which, in light of this gap, remains unproven. We note, however, that such a statement would follow from Lang’s conjecture that every bounded projective variety is in fact algebraically hyperbolic.