Curves in Hilbert modular varieties

TL;DR

This paper proves a boundedness theorem for abelian varieties with real multiplication, studies curves in Hilbert modular varieties, and confirms the Green-Griffiths-Lang conjecture for these varieties with few exceptions.

Contribution

It establishes the Green-Griffiths-Lang conjecture for Hilbert modular varieties using holomorphic foliations and metric methods, advancing understanding of entire curves in these spaces.

Findings

Boundedness theorem for families of abelian varieties with real multiplication

Second Main Theorem for entire curves in Hilbert modular varieties

Confirmation of the Green-Griffiths-Lang conjecture up to finitely many exceptions

Abstract

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.