Variétés faiblement spéciales à courbes entières dégénérées

Abstract

A complex projective manifold is said to be weakly special if none of its finite étale covers has a dominant rational map to a positive-dimensional manifold of general type. Weakly special manifolds are conjectured by Harris and Tschinkel (Rational points on quartics, Duke Math. J. 104 (2000), 477–500) to be potentially dense if defined over a number field. In a previous paper by F. Campana (Special varieties, orbifolds and classification theory, Ann. Inst. Fourier (Grenoble) 54 (2004), 499–665), the stronger notion of special (complex projective) manifolds was introduced. These are conjectured in (9.20) of that work to be exactly the potentially dense manifolds (when defined over a number field). The two notions of specialness and weak-specialness coincide up to dimension 2, but differ from dimension 3 on, as shown by examples $X$ constructed by Bogomolov and Tschinkel (Special elliptic fibrations, in Proc. Fanu Conf., Torino, 2003, ed. A. Conte), answering a question raised in Campana's paper. These examples should thus be potentially dense, according to the first conjecture, but should not be so, according to the second conjecture. The present techniques of arithmetic geometry do not seem to be able to decide between these two conflicting conjectures. So, instead of this, we investigate the more tractable hyperbolic aspects of the examples of Bogomolov and Tschinkel, and show that for certain manifolds $X$, the behaviour of entire transcendental curves is that given in Conjecture 9.16, p. 618 of Campana's above-mentioned work. More precisely, the image of any holomorphic map from ${\mathbb{C}}$ to $X$ is contained either in some fibre of the unique elliptic fibration on $X$, or in some fixed divisor of $X$. This is consistent with the standard links (Lang and Vojta for varieties of general type, the first author's previous work in general) expected between the hyperbolic and arithmetic behaviour of projective varieties in cases where the second conjecture (stated by Campana) holds. Conversely, if the conjecture stated by Harris and Tschinkel is true, these expected links between arithmetics and complex hyperbolicity should fail to hold on these examples already.