CURVES IN HILBERT MODULAR VARIETIES

Erwan Rousseau (1, 2) , Frédéric Touzet (3)

1 I2M - Institut de Mathématiques de Marseille
2 IUF - Institut universitaire de France
3 IRMAR - Institut de Recherche Mathématique de Rennes

We prove a boundedness-theorem for families of abelian varieties with real multiplication. More generally, we study curves in Hilbert modular varieties from the point of view of the Green Griffiths-Lang conjecture claiming that entire curves in complex projective varieties of general type should be contained in a proper subvariety. Using holomorphic foliations theory, we establish a Second Main Theorem following Nevanlinna theory. Finally, with a metric approach, we establish the strong Green-Griffiths-Lang conjecture for Hilbert modular varieties up to finitely many possible exceptions.

Domaines: Géométrie algébrique [math.AG]; Variables complexes [math.CV]

Erwan Rousseau, Frédéric Touzet. CURVES IN HILBERT MODULAR VARIETIES. Asian Journal of Mathematics, 2018, 22 (4), pp.673 - 690. ⟨10.4310/AJM.2018.v22.n4.a4⟩. ⟨hal-01103058v4⟩